(*** helper for §35 Step II: real-valued extension with range in [-1,1] ***)
(*** LATEX VERSION: Step 2 (core): extend f:A→(-1,1) by an infinite series of Urysohn functions; here packaged as existence of a continuous gR:X→R agreeing with f on A and mapping X into [-1,1]. ***)
(*** helper for §35 Step II: nonempty closed subset case, real-valued extension ***)
(*** LATEX VERSION: Step II (nonempty A): construct a real-valued continuous extension gR:X->R agreeing with f on A and bounded in [-1,1]. ***)
L173105
Let X, Tx, A and f be given.
L173106
Assume Hnorm: normal_space X Tx.
L173107
Assume HA: closed_in X Tx A.
L173108
Assume HAnemp: A ≠ Empty.
L173109
Assume Hf: continuous_map A (subspace_topology X Tx A) (closed_interval (minus_SNo 1) 1) (closed_interval_topology (minus_SNo 1) 1) f.
L173111
We will prove ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1).
L173115
Set I to be the term closed_interval (minus_SNo 1) 1.
L173117
We prove the intermediate claim HTx: topology_on X Tx.
(*** TeX Step II series construction: build gR as a uniformly convergent series of Urysohn-step functions (formalized below). ***)
L173119
An exact proof term for the current goal is (normal_space_topology_on X Tx Hnorm).
L173119
We prove the intermediate claim HAsubX: A ⊆ X.
L173121
An exact proof term for the current goal is (closed_in_subset X Tx A HA).
L173121
We prove the intermediate claim Hf_fun: function_on f A I.
L173123
An exact proof term for the current goal is (continuous_map_function_on A (subspace_topology X Tx A) I (closed_interval_topology (minus_SNo 1) 1) f Hf).
L173124
We prove the intermediate claim Hf_R: function_on f A R.
L173126
Let x be given.
L173126
Assume HxA: x ∈ A.
L173126
We prove the intermediate claim HfxI: apply_fun f x ∈ I.
L173128
An exact proof term for the current goal is (Hf_fun x HxA).
L173128
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f x) HfxI).
L173129
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
L173132
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
(*** TeX Step II outline: obtain g0 by Step I, then iterate on residuals and take a uniformly convergent series. ***)
L173133
We prove the intermediate claim Hexg0: ∃g0 : set, continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third).
L173141
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A f Hnorm HA Hf).
L173141
Apply Hexg0 to the current goal.
L173142
Let g0 be given.
L173143
Assume Hg0.
L173143
We prove the intermediate claim Hg0contI0: continuous_map X Tx I0 T0 g0.
L173145
We prove the intermediate claim Hleft: continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third).
L173149
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0).
L173155
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0) (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) Hleft).
L173160
We prove the intermediate claim Hg0contR: continuous_map X Tx R R_standard_topology g0.
L173162
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L173163
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology g0 Hg0contI0 (closed_interval_sub_R (minus_SNo one_third) one_third) R_standard_topology_is_topology_local HT0eq).
L173168
Set f1 to be the term graph A (λx : set ⇒ add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L173170
We prove the intermediate claim Hfung0: function_on g0 X I0.
(*** Residual on A: f1 = f - g0|A. ***)
L173172
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 g0 Hg0contI0).
L173172
We prove the intermediate claim Hf1_total: total_function_on f1 A R.
L173174
Apply (total_function_on_graph A R (λx : set ⇒ add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)))) to the current goal.
L173174
Let x be given.
L173175
Assume HxA: x ∈ A.
L173175
We will prove add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)) ∈ R.
L173176
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L173178
An exact proof term for the current goal is (Hf_R x HxA).
L173178
We prove the intermediate claim HgxI0: apply_fun g0 x ∈ I0.
L173180
An exact proof term for the current goal is (Hfung0 x (HAsubX x HxA)).
L173180
We prove the intermediate claim HgxR: apply_fun g0 x ∈ R.
L173182
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) HgxI0).
L173182
We prove the intermediate claim HmgxR: minus_SNo (apply_fun g0 x) ∈ R.
L173184
An exact proof term for the current goal is (real_minus_SNo (apply_fun g0 x) HgxR).
L173184
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo (apply_fun g0 x)) HmgxR).
L173185
We prove the intermediate claim Hg0pair: (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third).
L173193
An exact proof term for the current goal is Hg0.
L173193
We prove the intermediate claim Hg0_on_B: ∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third.
L173197
We prove the intermediate claim Hleft: continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third).
L173201
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0pair).
L173207
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 g0) (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third) Hleft).
L173212
We prove the intermediate claim Hg0_on_C: ∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third.
L173216
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 g0 ∧ (∀x : set, x ∈ preimage_of A f ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun g0 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f ((closed_interval one_third 1) ∩ I) → apply_fun g0 x = one_third) Hg0pair).
L173222
We prove the intermediate claim Hf1_apply: ∀x : set, x ∈ A → apply_fun f1 x = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L173226
Let x be given.
L173226
Assume HxA: x ∈ A.
L173226
rewrite the current goal using (apply_fun_graph A (λx0 : set ⇒ add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0))) x HxA) (from left to right).
Use reflexivity.
L173228
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L173230
Set I3 to be the term closed_interval one_third 1.
L173231
Set I2 to be the term closed_interval (minus_SNo two_thirds) two_thirds.
L173232
Set B to be the term preimage_of A f (I1 ∩ I).
L173233
Set C to be the term preimage_of A f (I3 ∩ I).
L173234
We prove the intermediate claim Hf1_range: ∀x : set, x ∈ A → apply_fun f1 x ∈ I2.
L173237
Let x be given.
L173237
Assume HxA: x ∈ A.
L173237
We prove the intermediate claim HfxI: apply_fun f x ∈ I.
L173239
An exact proof term for the current goal is (Hf_fun x HxA).
L173239
We prove the intermediate claim HxX: x ∈ X.
L173241
An exact proof term for the current goal is (HAsubX x HxA).
L173241
Apply (xm (x ∈ B)) to the current goal.
L173243
Assume HxB: x ∈ B.
L173243
We prove the intermediate claim Hg0eq: apply_fun g0 x = minus_SNo one_third.
L173245
An exact proof term for the current goal is (Hg0_on_B x HxB).
L173245
We prove the intermediate claim Hf1eq: apply_fun f1 x = add_SNo (apply_fun f x) one_third.
L173247
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L173247
rewrite the current goal using Hg0eq (from left to right) at position 1.
L173248
We prove the intermediate claim H13R: one_third ∈ R.
L173250
An exact proof term for the current goal is one_third_in_R.
L173250
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L173252
rewrite the current goal using Hf1eq (from left to right).
L173253
We prove the intermediate claim HfxI1I: apply_fun f x ∈ I1 ∩ I.
L173255
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f x0 ∈ I1 ∩ I) x HxB).
L173255
We prove the intermediate claim HfxI1: apply_fun f x ∈ I1.
L173257
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun f x) HfxI1I).
L173257
We prove the intermediate claim H13R: one_third ∈ R.
L173259
An exact proof term for the current goal is one_third_in_R.
L173259
We prove the intermediate claim H23R: two_thirds ∈ R.
L173261
An exact proof term for the current goal is two_thirds_in_R.
L173261
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L173263
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L173263
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L173265
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L173265
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L173267
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173267
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L173269
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) HfxI1).
L173269
We prove the intermediate claim Hfx_bounds: Rle (minus_SNo 1) (apply_fun f x) ∧ Rle (apply_fun f x) (minus_SNo one_third).
L173271
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) Hm1R Hm13R HfxI1).
L173272
We prove the intermediate claim Hm1lefx: Rle (minus_SNo 1) (apply_fun f x).
L173274
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) (minus_SNo one_third)) Hfx_bounds).
L173275
We prove the intermediate claim Hfxlem13: Rle (apply_fun f x) (minus_SNo one_third).
L173277
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) (minus_SNo one_third)) Hfx_bounds).
L173278
We prove the intermediate claim Hf1R: add_SNo (apply_fun f x) one_third ∈ R.
L173280
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR one_third H13R).
L173280
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun f x) one_third).
L173282
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun f x) one_third Hm1R HfxR H13R Hm1lefx).
L173282
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) one_third).
L173284
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L173284
An exact proof term for the current goal is Hlow_tmp.
L173285
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun f x) one_third) (add_SNo (minus_SNo one_third) one_third).
L173287
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) (minus_SNo one_third) one_third HfxR Hm13R H13R Hfxlem13).
L173287
We prove the intermediate claim H13S: SNo one_third.
L173289
An exact proof term for the current goal is (real_SNo one_third H13R).
L173289
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun f x) one_third) 0.
L173291
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L173291
An exact proof term for the current goal is Hup0_tmp.
L173292
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L173294
An exact proof term for the current goal is Rle_0_two_thirds.
L173294
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f x) one_third) two_thirds.
L173296
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f x) one_third) 0 two_thirds Hup0 H0le23).
L173296
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) one_third) Hm23R H23R Hf1R Hlow Hup).
L173299
Assume HxnotB: ¬ (x ∈ B).
L173299
Apply (xm (x ∈ C)) to the current goal.
L173301
Assume HxC: x ∈ C.
L173301
We prove the intermediate claim Hg0eq: apply_fun g0 x = one_third.
L173303
An exact proof term for the current goal is (Hg0_on_C x HxC).
L173303
We prove the intermediate claim Hf1eq: apply_fun f1 x = add_SNo (apply_fun f x) (minus_SNo one_third).
L173305
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L173305
rewrite the current goal using Hg0eq (from left to right) at position 1.
Use reflexivity.
L173307
rewrite the current goal using Hf1eq (from left to right).
L173308
We prove the intermediate claim HfxI3I: apply_fun f x ∈ I3 ∩ I.
L173310
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f x0 ∈ I3 ∩ I) x HxC).
L173310
We prove the intermediate claim HfxI3: apply_fun f x ∈ I3.
L173312
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun f x) HfxI3I).
L173312
We prove the intermediate claim H13R: one_third ∈ R.
L173314
An exact proof term for the current goal is one_third_in_R.
L173314
We prove the intermediate claim H23R: two_thirds ∈ R.
L173316
An exact proof term for the current goal is two_thirds_in_R.
L173316
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L173318
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L173318
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L173320
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173320
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L173322
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun f x) HfxI3).
L173322
We prove the intermediate claim Hfx_bounds: Rle one_third (apply_fun f x) ∧ Rle (apply_fun f x) 1.
L173324
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun f x) H13R real_1 HfxI3).
L173324
We prove the intermediate claim H13lefx: Rle one_third (apply_fun f x).
L173326
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds).
L173326
We prove the intermediate claim Hfxle1: Rle (apply_fun f x) 1.
L173328
An exact proof term for the current goal is (andER (Rle one_third (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds).
L173328
We prove the intermediate claim Hf1R: add_SNo (apply_fun f x) (minus_SNo one_third) ∈ R.
L173330
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo one_third) Hm13R).
L173330
We prove the intermediate claim H0le_f1_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L173333
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun f x) (minus_SNo one_third) H13R HfxR Hm13R H13lefx).
L173333
We prove the intermediate claim H13S: SNo one_third.
L173335
An exact proof term for the current goal is (real_SNo one_third H13R).
L173335
We prove the intermediate claim H0le_f1: Rle 0 (add_SNo (apply_fun f x) (minus_SNo one_third)).
L173337
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L173337
An exact proof term for the current goal is H0le_f1_tmp.
L173338
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L173340
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L173340
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L173342
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun f x) (minus_SNo one_third)) Hm23le0 H0le_f1).
L173342
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L173345
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) 1 (minus_SNo one_third) HfxR real_1 Hm13R Hfxle1).
L173345
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) two_thirds.
L173347
rewrite the current goal using two_thirds_eq_1_minus_one_third (from left to right) at position 1.
L173347
An exact proof term for the current goal is Hup_tmp.
L173348
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) (minus_SNo one_third)) Hm23R H23R Hf1R Hlow Hup).
L173352
Assume HxnotC: ¬ (x ∈ C).
L173352
We prove the intermediate claim HnotI1: ¬ (apply_fun f x ∈ I1).
L173354
Assume HfxI1: apply_fun f x ∈ I1.
L173354
We prove the intermediate claim HfxI1I: apply_fun f x ∈ I1 ∩ I.
L173356
An exact proof term for the current goal is (binintersectI I1 I (apply_fun f x) HfxI1 HfxI).
L173356
We prove the intermediate claim HxB': x ∈ B.
L173358
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f x0 ∈ I1 ∩ I) x HxA HfxI1I).
L173358
Apply FalseE to the current goal.
L173359
An exact proof term for the current goal is (HxnotB HxB').
L173360
We prove the intermediate claim HnotI3: ¬ (apply_fun f x ∈ I3).
L173362
Assume HfxI3: apply_fun f x ∈ I3.
L173362
We prove the intermediate claim HfxI3I: apply_fun f x ∈ I3 ∩ I.
L173364
An exact proof term for the current goal is (binintersectI I3 I (apply_fun f x) HfxI3 HfxI).
L173364
We prove the intermediate claim HxC': x ∈ C.
L173366
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f x0 ∈ I3 ∩ I) x HxA HfxI3I).
L173366
Apply FalseE to the current goal.
L173367
An exact proof term for the current goal is (HxnotC HxC').
L173368
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L173370
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f x) HfxI).
L173370
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L173372
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L173372
We prove the intermediate claim H13R: one_third ∈ R.
L173374
An exact proof term for the current goal is one_third_in_R.
L173374
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L173376
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173376
We prove the intermediate claim Hfx_bounds: Rle (minus_SNo 1) (apply_fun f x) ∧ Rle (apply_fun f x) 1.
L173378
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun f x) Hm1R real_1 HfxI).
L173378
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun f x) (minus_SNo 1)).
L173380
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun f x) (andEL (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds)).
L173381
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun f x)).
L173383
An exact proof term for the current goal is (RleE_nlt (apply_fun f x) 1 (andER (Rle (minus_SNo 1) (apply_fun f x)) (Rle (apply_fun f x) 1) Hfx_bounds)).
L173384
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun f x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun f x).
L173386
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun f x) Hm1R Hm13R HfxR HnotI1).
L173387
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun f x).
L173389
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun f x))) to the current goal.
L173390
Assume Hbad: Rlt (apply_fun f x) (minus_SNo 1).
L173390
Apply FalseE to the current goal.
L173391
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L173393
Assume Hok: Rlt (minus_SNo one_third) (apply_fun f x).
L173393
An exact proof term for the current goal is Hok.
L173394
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun f x) (minus_SNo one_third)).
L173396
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun f x) Hm13lt_fx).
L173396
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun f x) one_third ∨ Rlt 1 (apply_fun f x).
L173398
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun f x) H13R real_1 HfxR HnotI3).
L173399
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun f x) one_third.
L173401
Apply (HnotI3_cases (Rlt (apply_fun f x) one_third)) to the current goal.
L173402
Assume Hok: Rlt (apply_fun f x) one_third.
L173402
An exact proof term for the current goal is Hok.
L173404
Assume Hbad: Rlt 1 (apply_fun f x).
L173404
Apply FalseE to the current goal.
L173405
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L173406
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun f x)).
L173408
An exact proof term for the current goal is (not_Rlt_sym (apply_fun f x) one_third Hfx_lt_13).
L173408
We prove the intermediate claim HfxI0: apply_fun f x ∈ I0.
L173410
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L173412
We prove the intermediate claim HxSep: apply_fun f x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L173414
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun f x) HfxR (andI (¬ (Rlt (apply_fun f x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun f x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L173418
rewrite the current goal using HI0_def (from left to right).
L173419
An exact proof term for the current goal is HxSep.
L173420
rewrite the current goal using (Hf1_apply x HxA) (from left to right).
L173421
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
L173423
An exact proof term for the current goal is (Hfung0 x HxX).
L173423
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L173425
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0xI0).
L173425
We prove the intermediate claim Hm_g0x_R: minus_SNo (apply_fun g0 x) ∈ R.
L173427
An exact proof term for the current goal is (real_minus_SNo (apply_fun g0 x) Hg0xR).
L173427
We prove the intermediate claim Hf1xR: add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)) ∈ R.
L173429
An exact proof term for the current goal is (real_add_SNo (apply_fun f x) HfxR (minus_SNo (apply_fun g0 x)) Hm_g0x_R).
L173429
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L173431
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173431
We prove the intermediate claim H23R: two_thirds ∈ R.
L173433
An exact proof term for the current goal is two_thirds_in_R.
L173433
We prove the intermediate claim Hm23R: (minus_SNo two_thirds) ∈ R.
L173435
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L173435
We prove the intermediate claim Hfx0_bounds: Rle (minus_SNo one_third) (apply_fun f x) ∧ Rle (apply_fun f x) one_third.
L173437
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun f x) Hm13R H13R HfxI0).
L173437
We prove the intermediate claim Hg0x_bounds: Rle (minus_SNo one_third) (apply_fun g0 x) ∧ Rle (apply_fun g0 x) one_third.
L173439
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun g0 x) Hm13R H13R Hg0xI0).
L173439
We prove the intermediate claim Hm13_le_fx: Rle (minus_SNo one_third) (apply_fun f x).
L173441
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun f x)) (Rle (apply_fun f x) one_third) Hfx0_bounds).
L173441
We prove the intermediate claim Hfx_le_13: Rle (apply_fun f x) one_third.
L173443
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun f x)) (Rle (apply_fun f x) one_third) Hfx0_bounds).
L173443
We prove the intermediate claim Hm13_le_g0x: Rle (minus_SNo one_third) (apply_fun g0 x).
L173445
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun g0 x)) (Rle (apply_fun g0 x) one_third) Hg0x_bounds).
L173445
We prove the intermediate claim Hg0x_le_13: Rle (apply_fun g0 x) one_third.
L173447
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun g0 x)) (Rle (apply_fun g0 x) one_third) Hg0x_bounds).
L173447
We prove the intermediate claim Hm13_le_mg0x: Rle (minus_SNo one_third) (minus_SNo (apply_fun g0 x)).
L173449
An exact proof term for the current goal is (Rle_minus_contra (apply_fun g0 x) one_third Hg0x_le_13).
L173449
We prove the intermediate claim Hmg0x_le_13: Rle (minus_SNo (apply_fun g0 x)) one_third.
L173451
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun g0 x)) (minus_SNo (minus_SNo one_third)).
L173452
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun g0 x) Hm13_le_g0x).
L173452
We prove the intermediate claim H13R: one_third ∈ R.
L173454
An exact proof term for the current goal is one_third_in_R.
L173454
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L173455
An exact proof term for the current goal is Htmp.
L173456
We prove the intermediate claim Hlo1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)).
L173459
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun f x) (minus_SNo one_third) Hm13R HfxR Hm13R Hm13_le_fx).
L173460
We prove the intermediate claim Hlo2: Rle (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L173463
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun f x) (minus_SNo one_third) (minus_SNo (apply_fun g0 x)) HfxR Hm13R Hm_g0x_R Hm13_le_mg0x).
L173464
We prove the intermediate claim Hlo': Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L173467
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo one_third)) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) Hlo1 Hlo2).
L173470
We prove the intermediate claim Hlo: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L173472
rewrite the current goal using (minus_two_thirds_eq) (from left to right) at position 1.
L173472
An exact proof term for the current goal is Hlo'.
L173473
We prove the intermediate claim Hhi1: Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third (minus_SNo (apply_fun g0 x))).
L173476
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f x) one_third (minus_SNo (apply_fun g0 x)) HfxR H13R Hm_g0x_R Hfx_le_13).
L173477
We prove the intermediate claim Hhi2: Rle (add_SNo one_third (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third).
L173480
An exact proof term for the current goal is (Rle_add_SNo_2 one_third (minus_SNo (apply_fun g0 x)) one_third H13R Hm_g0x_R H13R Hmg0x_le_13).
L173481
We prove the intermediate claim Hhi': Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third).
L173484
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) (add_SNo one_third (minus_SNo (apply_fun g0 x))) (add_SNo one_third one_third) Hhi1 Hhi2).
L173487
We prove the intermediate claim Hhi: Rle (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) two_thirds.
L173489
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L173490
rewrite the current goal using Hdef23 (from left to right).
L173491
An exact proof term for the current goal is Hhi'.
L173492
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))) Hm23R H23R Hf1xR Hlo Hhi).
L173495
We prove the intermediate claim Hseries: ∃gR : set, continuous_map X Tx R R_standard_topology gR ∧ (∀x : set, x ∈ A → apply_fun gR x = apply_fun f x) ∧ (∀x : set, x ∈ X → apply_fun gR x ∈ closed_interval (minus_SNo 1) 1).
(*** Further iteration and summation still pending. ***)
L173502
Set I to be the term closed_interval (minus_SNo 1) 1.
L173503
Set Ti to be the term closed_interval_topology (minus_SNo 1) 1.
(*** TeX Step II: geometric series of successive Step I corrections. ***)
L173504
We prove the intermediate claim H23R: two_thirds ∈ R.
L173506
An exact proof term for the current goal is two_thirds_in_R.
L173506
We prove the intermediate claim H23S: SNo two_thirds.
L173508
An exact proof term for the current goal is (real_SNo two_thirds H23R).
L173508
We prove the intermediate claim H23pos: 0 < two_thirds.
L173510
An exact proof term for the current goal is two_thirds_pos.
L173510
We prove the intermediate claim H23ne0: two_thirds ≠ 0.
L173512
An exact proof term for the current goal is two_thirds_ne0.
L173512
Set den to be the term two_thirds.
L173515
Set f1s to be the term compose_fun A f1 (div_const_fun den).
(*** define scaled residual on A: f1s = f1 / (2/3), so f1s maps into [-1,1] ***)
L173516
We prove the intermediate claim Hdivfun: function_on (div_const_fun den) R R.
L173518
Let t be given.
L173518
Assume HtR: t ∈ R.
L173518
An exact proof term for the current goal is (div_const_fun_value_in_R den t H23R HtR).
L173519
We prove the intermediate claim Hf1fun0: function_on f1 A R.
L173521
An exact proof term for the current goal is (andEL (function_on f1 A R) (∀a : set, a ∈ A → ∃y : set, y ∈ R ∧ (a,y) ∈ f1) Hf1_total).
L173523
We prove the intermediate claim Hf1s_total: total_function_on f1s A R.
L173525
An exact proof term for the current goal is (total_function_on_compose_fun A R R f1 (div_const_fun den) Hf1fun0 Hdivfun).
L173525
We prove the intermediate claim Hf1s_fun: function_on f1s A R.
L173527
An exact proof term for the current goal is (andEL (function_on f1s A R) (∀x : set, x ∈ A → ∃y : set, y ∈ R ∧ (x,y) ∈ f1s) Hf1s_total).
L173529
We prove the intermediate claim Hf1s_apply: ∀x : set, x ∈ A → apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L173532
Let x be given.
L173532
Assume HxA: x ∈ A.
L173532
We prove the intermediate claim HxR: apply_fun f1 x ∈ R.
L173534
An exact proof term for the current goal is (Hf1fun0 x HxA).
L173534
rewrite the current goal using (compose_fun_apply A f1 (div_const_fun den) x HxA) (from left to right).
L173535
rewrite the current goal using (div_const_fun_apply den (apply_fun f1 x) H23R HxR) (from left to right).
Use reflexivity.
L173537
We prove the intermediate claim Hf1s_I: ∀x : set, x ∈ A → apply_fun f1s x ∈ I.
(*** pending: prove f1s is continuous as a map A -> I with the interval topology ***)
L173541
Let x be given.
L173541
Assume HxA: x ∈ A.
L173541
We will prove apply_fun f1s x ∈ I.
L173542
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L173544
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L173544
We prove the intermediate claim Hf1xI2: apply_fun f1 x ∈ I2.
L173547
An exact proof term for the current goal is (Hf1_range x HxA).
L173547
We prove the intermediate claim Hm23R: (minus_SNo den) ∈ R.
L173549
An exact proof term for the current goal is (real_minus_SNo den H23R).
L173549
We prove the intermediate claim Hf1xR: apply_fun f1 x ∈ R.
L173551
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun f1 x) Hf1xI2).
L173551
We prove the intermediate claim Hf1xS: SNo (apply_fun f1 x).
L173553
An exact proof term for the current goal is (real_SNo (apply_fun f1 x) Hf1xR).
L173553
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun f1 x) ∧ Rle (apply_fun f1 x) den.
L173556
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun f1 x) Hm23R H23R Hf1xI2).
L173556
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun f1 x).
L173558
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun f1 x)) (Rle (apply_fun f1 x) den) Hbounds).
L173560
We prove the intermediate claim Hhi: Rle (apply_fun f1 x) den.
L173562
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun f1 x)) (Rle (apply_fun f1 x) den) Hbounds).
L173564
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun f1 x)).
L173566
An exact proof term for the current goal is (RleE_nlt (apply_fun f1 x) den Hhi).
L173566
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun f1 x) (minus_SNo den)).
L173568
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun f1 x) Hlo).
L173568
We prove the intermediate claim HyEq: apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L173571
An exact proof term for the current goal is (Hf1s_apply x HxA).
L173571
We prove the intermediate claim HyR: apply_fun f1s x ∈ R.
L173573
rewrite the current goal using HyEq (from left to right).
L173573
An exact proof term for the current goal is (real_div_SNo (apply_fun f1 x) Hf1xR den H23R).
L173574
We prove the intermediate claim HyS: SNo (apply_fun f1s x).
L173576
An exact proof term for the current goal is (real_SNo (apply_fun f1s x) HyR).
L173576
We prove the intermediate claim Hy_le_1: Rle (apply_fun f1s x) 1.
L173579
Apply (RleI (apply_fun f1s x) 1 HyR real_1) to the current goal.
L173579
We will prove ¬ (Rlt 1 (apply_fun f1s x)).
L173580
Assume H1lt: Rlt 1 (apply_fun f1s x).
L173581
We prove the intermediate claim H1lty: 1 < apply_fun f1s x.
L173583
An exact proof term for the current goal is (RltE_lt 1 (apply_fun f1s x) H1lt).
L173583
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun f1s x) den.
L173585
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun f1s x) den SNo_1 HyS H23S H23pos H1lty).
L173585
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun f1s x) den.
L173587
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L173587
An exact proof term for the current goal is HmulLt.
L173588
We prove the intermediate claim HmulEq: mul_SNo (apply_fun f1s x) den = apply_fun f1 x.
L173590
rewrite the current goal using HyEq (from left to right) at position 1.
L173590
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L173591
We prove the intermediate claim Hden_lt_f1x: den < apply_fun f1 x.
L173593
rewrite the current goal using HmulEq (from right to left).
L173593
An exact proof term for the current goal is HmulLt'.
L173594
We prove the intermediate claim Hbad: Rlt den (apply_fun f1 x).
L173596
An exact proof term for the current goal is (RltI den (apply_fun f1 x) H23R Hf1xR Hden_lt_f1x).
L173596
An exact proof term for the current goal is (Hnlt_hi Hbad).
L173597
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun f1s x).
L173600
Apply (RleI (minus_SNo 1) (apply_fun f1s x) Hm1R HyR) to the current goal.
L173600
We will prove ¬ (Rlt (apply_fun f1s x) (minus_SNo 1)).
L173601
Assume Hylt: Rlt (apply_fun f1s x) (minus_SNo 1).
L173602
We prove the intermediate claim Hylts: apply_fun f1s x < minus_SNo 1.
L173604
An exact proof term for the current goal is (RltE_lt (apply_fun f1s x) (minus_SNo 1) Hylt).
L173604
We prove the intermediate claim HmulLt: mul_SNo (apply_fun f1s x) den < mul_SNo (minus_SNo 1) den.
L173606
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun f1s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L173607
We prove the intermediate claim HmulEq: mul_SNo (apply_fun f1s x) den = apply_fun f1 x.
L173609
rewrite the current goal using HyEq (from left to right) at position 1.
L173609
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L173610
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L173612
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L173612
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L173614
We prove the intermediate claim Hf1x_lt_mden: apply_fun f1 x < minus_SNo den.
L173616
rewrite the current goal using HmulEq (from right to left).
L173616
rewrite the current goal using HrhsEq (from right to left).
L173617
An exact proof term for the current goal is HmulLt.
L173618
We prove the intermediate claim Hbad: Rlt (apply_fun f1 x) (minus_SNo den).
L173620
An exact proof term for the current goal is (RltI (apply_fun f1 x) (minus_SNo den) Hf1xR Hm23R Hf1x_lt_mden).
L173620
An exact proof term for the current goal is (Hnlt_lo Hbad).
L173621
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun f1s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L173623
We prove the intermediate claim Hf1s_cont: continuous_map A (subspace_topology X Tx A) I Ti f1s.
L173625
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
(*** reduce to continuity into R and range restriction to I ***)
L173627
An exact proof term for the current goal is R_standard_topology_is_topology.
L173627
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L173629
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L173629
We prove the intermediate claim Hf1cont: continuous_map A (subspace_topology X Tx A) R R_standard_topology f1.
L173631
Set Ta to be the term subspace_topology X Tx A.
L173632
We prove the intermediate claim HTa: topology_on A Ta.
(*** f1 is composition of pair_map with add_fun_R ***)
L173634
An exact proof term for the current goal is (subspace_topology_is_topology X Tx A HTx HAsubX).
L173634
We prove the intermediate claim HIcR: I ⊆ R.
L173636
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L173636
We prove the intermediate claim HTiEq: (closed_interval_topology (minus_SNo 1) 1) = subspace_topology R R_standard_topology I.
Use reflexivity.
L173639
We prove the intermediate claim HfcontR: continuous_map A Ta R R_standard_topology f.
L173641
An exact proof term for the current goal is (continuous_map_range_expand A Ta I (closed_interval_topology (minus_SNo 1) 1) R R_standard_topology f Hf HIcR R_standard_topology_is_topology_local HTiEq).
L173646
We prove the intermediate claim Hg0contA: continuous_map A Ta R R_standard_topology g0.
L173648
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology g0 A HTx HAsubX Hg0contR).
L173648
We prove the intermediate claim Hnegcont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L173650
An exact proof term for the current goal is neg_fun_continuous.
L173650
Set g0neg to be the term compose_fun A g0 neg_fun.
L173651
We prove the intermediate claim Hg0negcont: continuous_map A Ta R R_standard_topology g0neg.
L173653
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology g0 neg_fun Hg0contA Hnegcont).
L173654
Set h to be the term pair_map A f g0neg.
L173655
We prove the intermediate claim Hhcont: continuous_map A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h.
L173659
An exact proof term for the current goal is (maps_into_products_axiom A Ta R R_standard_topology R R_standard_topology f g0neg HfcontR Hg0negcont).
L173660
We prove the intermediate claim Haddcont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L173665
An exact proof term for the current goal is add_fun_R_continuous.
L173665
Set f1c to be the term compose_fun A h add_fun_R.
L173666
We prove the intermediate claim Hf1c_cont: continuous_map A Ta R R_standard_topology f1c.
L173668
An exact proof term for the current goal is (composition_continuous A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology h add_fun_R Hhcont Haddcont).
L173671
We prove the intermediate claim Heq: f1 = f1c.
L173673
Apply set_ext to the current goal.
L173674
Let p be given.
L173674
Assume Hp: p ∈ f1.
L173674
We will prove p ∈ f1c.
L173675
Apply (ReplE_impred A (λx0 : set ⇒ (x0,add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0)))) p Hp (p ∈ f1c)) to the current goal.
L173680
Let x be given.
L173681
Assume HxA: x ∈ A.
L173681
Assume Hpeq: p = (x,add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x))).
L173682
rewrite the current goal using Hpeq (from left to right).
L173683
We prove the intermediate claim Hhx: apply_fun h x = (apply_fun f x,apply_fun g0neg x).
L173685
An exact proof term for the current goal is (pair_map_apply A R R f g0neg x HxA).
L173685
We prove the intermediate claim Hg0negx: apply_fun g0neg x = minus_SNo (apply_fun g0 x).
L173687
We prove the intermediate claim HxTa: x ∈ A.
L173688
An exact proof term for the current goal is HxA.
L173688
We prove the intermediate claim HxX: x ∈ X.
L173690
An exact proof term for the current goal is (HAsubX x HxA).
L173690
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L173692
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR x HxX).
L173692
rewrite the current goal using (compose_fun_apply A g0 neg_fun x HxA) (from left to right) at position 1.
L173693
rewrite the current goal using (neg_fun_apply (apply_fun g0 x) Hg0xR) (from left to right) at position 1.
Use reflexivity.
L173695
We prove the intermediate claim Hhimg: apply_fun h x ∈ setprod R R.
L173697
An exact proof term for the current goal is (continuous_map_function_on A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h Hhcont x HxA).
L173698
We prove the intermediate claim Happ: apply_fun f1c x = apply_fun add_fun_R (apply_fun h x).
L173700
An exact proof term for the current goal is (compose_fun_apply A h add_fun_R x HxA).
L173700
We prove the intermediate claim Hadd: apply_fun add_fun_R (apply_fun h x) = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L173703
rewrite the current goal using (add_fun_R_apply (apply_fun h x) Hhimg) (from left to right) at position 1.
L173703
rewrite the current goal using Hhx (from left to right).
L173704
rewrite the current goal using (tuple_2_0_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L173705
rewrite the current goal using (tuple_2_1_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L173706
rewrite the current goal using Hg0negx (from left to right) at position 1.
Use reflexivity.
L173708
rewrite the current goal using Hadd (from right to left).
L173709
An exact proof term for the current goal is (ReplI A (λx0 : set ⇒ (x0,apply_fun add_fun_R (apply_fun h x0))) x HxA).
L173711
Let p be given.
L173711
Assume Hp: p ∈ f1c.
L173711
We will prove p ∈ f1.
L173712
Apply (ReplE_impred A (λx0 : set ⇒ (x0,apply_fun add_fun_R (apply_fun h x0))) p Hp (p ∈ f1)) to the current goal.
L173713
Let x be given.
L173714
Assume HxA: x ∈ A.
L173714
Assume Hpeq: p = (x,apply_fun add_fun_R (apply_fun h x)).
L173715
rewrite the current goal using Hpeq (from left to right).
L173716
We prove the intermediate claim Hhx: apply_fun h x = (apply_fun f x,apply_fun g0neg x).
L173718
An exact proof term for the current goal is (pair_map_apply A R R f g0neg x HxA).
L173718
We prove the intermediate claim Hg0xR: apply_fun g0 x ∈ R.
L173720
We prove the intermediate claim HxX: x ∈ X.
L173721
An exact proof term for the current goal is (HAsubX x HxA).
L173721
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR x HxX).
L173722
We prove the intermediate claim Hg0negx: apply_fun g0neg x = minus_SNo (apply_fun g0 x).
L173724
rewrite the current goal using (compose_fun_apply A g0 neg_fun x HxA) (from left to right) at position 1.
L173724
rewrite the current goal using (neg_fun_apply (apply_fun g0 x) Hg0xR) (from left to right) at position 1.
Use reflexivity.
L173726
We prove the intermediate claim Hadd: apply_fun add_fun_R (apply_fun h x) = add_SNo (apply_fun f x) (minus_SNo (apply_fun g0 x)).
L173729
We prove the intermediate claim Hhimg: apply_fun h x ∈ setprod R R.
L173730
An exact proof term for the current goal is (continuous_map_function_on A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h Hhcont x HxA).
L173731
rewrite the current goal using (add_fun_R_apply (apply_fun h x) Hhimg) (from left to right) at position 1.
L173732
rewrite the current goal using Hhx (from left to right).
L173733
rewrite the current goal using (tuple_2_0_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L173734
rewrite the current goal using (tuple_2_1_eq (apply_fun f x) (apply_fun g0neg x)) (from left to right).
L173735
rewrite the current goal using Hg0negx (from left to right) at position 1.
Use reflexivity.
L173737
rewrite the current goal using Hadd (from left to right).
L173738
An exact proof term for the current goal is (ReplI A (λx0 : set ⇒ (x0,add_SNo (apply_fun f x0) (minus_SNo (apply_fun g0 x0)))) x HxA).
L173742
rewrite the current goal using Heq (from left to right).
L173743
An exact proof term for the current goal is Hf1c_cont.
L173744
We prove the intermediate claim Hf1s_cont_R: continuous_map A (subspace_topology X Tx A) R R_standard_topology f1s.
L173746
An exact proof term for the current goal is (composition_continuous A (subspace_topology X Tx A) R R_standard_topology R R_standard_topology f1 (div_const_fun den) Hf1cont Hdivcont).
L173747
We prove the intermediate claim HISubR: I ⊆ R.
L173749
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L173749
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L173751
rewrite the current goal using HTiEq (from left to right).
L173752
An exact proof term for the current goal is (continuous_map_range_restrict A (subspace_topology X Tx A) R R_standard_topology f1s I Hf1s_cont_R HISubR Hf1s_I).
L173754
We prove the intermediate claim Hex_u1: ∃u1 : set, continuous_map X Tx (closed_interval (minus_SNo one_third) one_third) (closed_interval_topology (minus_SNo one_third) one_third) u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third).
(*** apply Step I to f1s to obtain the next correction u1:X -> [-1/3,1/3] ***)
L173764
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A f1s Hnorm HA Hf1s_cont).
L173764
Apply Hex_u1 to the current goal.
L173765
Let u1 be given.
L173766
Assume Hu1.
L173766
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
L173767
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
L173768
We prove the intermediate claim Hu1AB: continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third).
L173773
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third) Hu1).
L173779
We prove the intermediate claim Hu1contI0: continuous_map X Tx I0 T0 u1.
L173781
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u1) (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) Hu1AB).
L173785
We prove the intermediate claim Hu1contR: continuous_map X Tx R R_standard_topology u1.
L173787
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L173788
We prove the intermediate claim HI0subR: I0 ⊆ R.
L173790
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L173790
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u1 Hu1contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L173798
Set den to be the term two_thirds.
L173800
Set u1s to be the term compose_fun X u1 (mul_const_fun den).
(*** a first partial sum g1 = g0 + (2/3) u1 is continuous ***)
L173801
We prove the intermediate claim HdenR: den ∈ R.
L173803
An exact proof term for the current goal is two_thirds_in_R.
L173803
We prove the intermediate claim HdenPos: 0 < den.
L173805
An exact proof term for the current goal is two_thirds_pos.
L173805
We prove the intermediate claim HmulCont: continuous_map R R_standard_topology R R_standard_topology (mul_const_fun den).
L173807
An exact proof term for the current goal is (mul_const_fun_continuous_pos den HdenR HdenPos).
L173807
We prove the intermediate claim Hu1s_cont: continuous_map X Tx R R_standard_topology u1s.
L173809
An exact proof term for the current goal is (composition_continuous X Tx R R_standard_topology R R_standard_topology u1 (mul_const_fun den) Hu1contR HmulCont).
L173810
Set h1 to be the term pair_map X g0 u1s.
L173811
We prove the intermediate claim Hh1cont: continuous_map X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) h1.
L173815
An exact proof term for the current goal is (maps_into_products_axiom X Tx R R_standard_topology R R_standard_topology g0 u1s Hg0contR Hu1s_cont).
L173816
We prove the intermediate claim Haddcont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L173821
An exact proof term for the current goal is add_fun_R_continuous.
L173821
Set g1 to be the term compose_fun X h1 add_fun_R.
L173822
We prove the intermediate claim Hg1cont: continuous_map X Tx R R_standard_topology g1.
L173824
An exact proof term for the current goal is (composition_continuous X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology h1 add_fun_R Hh1cont Haddcont).
L173827
Set Ta to be the term subspace_topology X Tx A.
L173830
We prove the intermediate claim HTa: topology_on A Ta.
(*** second correction step: build the next residual on A and a second partial sum ***)
L173832
An exact proof term for the current goal is (subspace_topology_is_topology X Tx A HTx HAsubX).
L173832
We prove the intermediate claim HIcR: I ⊆ R.
L173834
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L173834
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L173836
We prove the intermediate claim Hf1s_contR: continuous_map A Ta R R_standard_topology f1s.
L173838
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology f1s Hf1s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L173839
We prove the intermediate claim Hu1contA: continuous_map A Ta R R_standard_topology u1.
L173841
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u1 A HTx HAsubX Hu1contR).
L173841
We prove the intermediate claim Hnegcont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L173843
An exact proof term for the current goal is neg_fun_continuous.
L173843
Set u1neg to be the term compose_fun A u1 neg_fun.
L173844
We prove the intermediate claim Hu1neg_cont: continuous_map A Ta R R_standard_topology u1neg.
L173846
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u1 neg_fun Hu1contA Hnegcont).
L173847
Set r1 to be the term compose_fun A (pair_map A f1s u1neg) add_fun_R.
L173848
We prove the intermediate claim Hr1_cont: continuous_map A Ta R R_standard_topology r1.
L173850
An exact proof term for the current goal is (add_two_continuous_R A Ta f1s u1neg HTa Hf1s_contR Hu1neg_cont).
L173850
We prove the intermediate claim Hr1_apply: ∀x : set, x ∈ A → apply_fun r1 x = add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x)).
L173853
Let x be given.
L173853
Assume HxA: x ∈ A.
L173853
We prove the intermediate claim Hpimg: apply_fun (pair_map A f1s u1neg) x ∈ setprod R R.
L173855
rewrite the current goal using (pair_map_apply A R R f1s u1neg x HxA) (from left to right).
L173855
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L173857
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology f1s Hf1s_contR x HxA).
L173857
We prove the intermediate claim Hu1negRx: apply_fun u1neg x ∈ R.
L173859
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u1neg Hu1neg_cont x HxA).
L173859
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun f1s x) (apply_fun u1neg x) Hf1sRx Hu1negRx).
L173860
rewrite the current goal using (compose_fun_apply A (pair_map A f1s u1neg) add_fun_R x HxA) (from left to right).
L173861
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A f1s u1neg) x) Hpimg) (from left to right) at position 1.
L173862
rewrite the current goal using (pair_map_apply A R R f1s u1neg x HxA) (from left to right).
L173863
rewrite the current goal using (tuple_2_0_eq (apply_fun f1s x) (apply_fun u1neg x)) (from left to right).
L173864
rewrite the current goal using (tuple_2_1_eq (apply_fun f1s x) (apply_fun u1neg x)) (from left to right).
L173865
rewrite the current goal using (compose_fun_apply A u1 neg_fun x HxA) (from left to right) at position 1.
L173866
We prove the intermediate claim Hu1Rx: apply_fun u1 x ∈ R.
L173868
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u1 Hu1contA x HxA).
L173868
rewrite the current goal using (neg_fun_apply (apply_fun u1 x) Hu1Rx) (from left to right) at position 1.
Use reflexivity.
L173870
We prove the intermediate claim Hu1_on_B1: ∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third.
(*** show the next residual stays within [-2/3,2/3] on A ***)
L173876
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u1) (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third) Hu1AB).
L173880
We prove the intermediate claim Hu1_on_C1: ∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third.
L173884
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u1 ∧ (∀x : set, x ∈ preimage_of A f1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u1 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A f1s ((closed_interval one_third 1) ∩ I) → apply_fun u1 x = one_third) Hu1).
L173890
We prove the intermediate claim Hr1_range: ∀x : set, x ∈ A → apply_fun r1 x ∈ I2.
L173892
Let x be given.
L173892
Assume HxA: x ∈ A.
L173892
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L173893
Set I3 to be the term closed_interval one_third 1.
L173894
Set B1 to be the term preimage_of A f1s (I1 ∩ I).
L173895
Set C1 to be the term preimage_of A f1s (I3 ∩ I).
L173896
We prove the intermediate claim Hf1sIx: apply_fun f1s x ∈ I.
L173898
An exact proof term for the current goal is (Hf1s_I x HxA).
L173898
We prove the intermediate claim HB1_cases: x ∈ B1 ∨ ¬ (x ∈ B1).
L173900
An exact proof term for the current goal is (xm (x ∈ B1)).
L173900
Apply (HB1_cases (apply_fun r1 x ∈ I2)) to the current goal.
L173902
Assume HxB1: x ∈ B1.
L173902
We prove the intermediate claim Hu1eq: apply_fun u1 x = minus_SNo one_third.
L173904
An exact proof term for the current goal is (Hu1_on_B1 x HxB1).
L173904
We prove the intermediate claim Hr1eq: apply_fun r1 x = add_SNo (apply_fun f1s x) one_third.
L173906
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L173906
rewrite the current goal using Hu1eq (from left to right) at position 1.
L173907
We prove the intermediate claim H13R: one_third ∈ R.
L173909
An exact proof term for the current goal is one_third_in_R.
L173909
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L173911
rewrite the current goal using Hr1eq (from left to right).
L173912
We prove the intermediate claim Hf1sI1I: apply_fun f1s x ∈ I1 ∩ I.
L173914
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f1s x0 ∈ I1 ∩ I) x HxB1).
L173914
We prove the intermediate claim Hf1sI1: apply_fun f1s x ∈ I1.
L173916
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun f1s x) Hf1sI1I).
L173916
We prove the intermediate claim H13R: one_third ∈ R.
L173918
An exact proof term for the current goal is one_third_in_R.
L173918
We prove the intermediate claim H23R: two_thirds ∈ R.
L173920
An exact proof term for the current goal is two_thirds_in_R.
L173920
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L173922
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L173922
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L173924
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L173924
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L173926
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173926
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L173928
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hf1sI1).
L173928
We prove the intermediate claim Hf1s_bounds: Rle (minus_SNo 1) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) (minus_SNo one_third).
L173930
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hm1R Hm13R Hf1sI1).
L173931
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun f1s x).
L173933
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) (minus_SNo one_third)) Hf1s_bounds).
L173934
We prove the intermediate claim HhiI1: Rle (apply_fun f1s x) (minus_SNo one_third).
L173936
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) (minus_SNo one_third)) Hf1s_bounds).
L173937
We prove the intermediate claim Hr1Rx: add_SNo (apply_fun f1s x) one_third ∈ R.
L173939
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) Hf1sRx one_third H13R).
L173939
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun f1s x) one_third).
L173941
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun f1s x) one_third Hm1R Hf1sRx H13R Hm1le).
L173941
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) one_third).
L173943
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L173943
An exact proof term for the current goal is Hlow_tmp.
L173944
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun f1s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L173946
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) (minus_SNo one_third) one_third Hf1sRx Hm13R H13R HhiI1).
L173946
We prove the intermediate claim H13S: SNo one_third.
L173948
An exact proof term for the current goal is (real_SNo one_third H13R).
L173948
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun f1s x) one_third) 0.
L173950
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L173950
An exact proof term for the current goal is Hup0_tmp.
L173951
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L173953
An exact proof term for the current goal is Rle_0_two_thirds.
L173953
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) one_third) two_thirds.
L173955
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) one_third) 0 two_thirds Hup0 H0le23).
L173955
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) one_third) Hm23R H23R Hr1Rx Hlow Hup).
L173959
Assume HxnotB1: ¬ (x ∈ B1).
L173959
We prove the intermediate claim HC1_cases: x ∈ C1 ∨ ¬ (x ∈ C1).
L173961
An exact proof term for the current goal is (xm (x ∈ C1)).
L173961
Apply (HC1_cases (apply_fun r1 x ∈ I2)) to the current goal.
L173963
Assume HxC1: x ∈ C1.
L173963
We prove the intermediate claim Hu1eq: apply_fun u1 x = one_third.
L173965
An exact proof term for the current goal is (Hu1_on_C1 x HxC1).
L173965
We prove the intermediate claim Hr1eq: apply_fun r1 x = add_SNo (apply_fun f1s x) (minus_SNo one_third).
L173967
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L173967
rewrite the current goal using Hu1eq (from left to right) at position 1.
Use reflexivity.
L173969
rewrite the current goal using Hr1eq (from left to right).
L173970
We prove the intermediate claim Hf1sI3I: apply_fun f1s x ∈ I3 ∩ I.
L173972
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun f1s x0 ∈ I3 ∩ I) x HxC1).
L173972
We prove the intermediate claim Hf1sI3: apply_fun f1s x ∈ I3.
L173974
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun f1s x) Hf1sI3I).
L173974
We prove the intermediate claim H13R: one_third ∈ R.
L173976
An exact proof term for the current goal is one_third_in_R.
L173976
We prove the intermediate claim H23R: two_thirds ∈ R.
L173978
An exact proof term for the current goal is two_thirds_in_R.
L173978
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L173980
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L173980
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L173982
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L173982
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L173984
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun f1s x) Hf1sI3).
L173984
We prove the intermediate claim Hf1s_bounds: Rle one_third (apply_fun f1s x) ∧ Rle (apply_fun f1s x) 1.
L173986
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun f1s x) H13R real_1 Hf1sI3).
L173986
We prove the intermediate claim HloI3: Rle one_third (apply_fun f1s x).
L173988
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_bounds).
L173988
We prove the intermediate claim HhiI3: Rle (apply_fun f1s x) 1.
L173990
An exact proof term for the current goal is (andER (Rle one_third (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_bounds).
L173990
We prove the intermediate claim Hr1Rx: add_SNo (apply_fun f1s x) (minus_SNo one_third) ∈ R.
L173992
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) Hf1sRx (minus_SNo one_third) Hm13R).
L173992
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L173995
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun f1s x) (minus_SNo one_third) H13R Hf1sRx Hm13R HloI3).
L173996
We prove the intermediate claim H13S: SNo one_third.
L173998
An exact proof term for the current goal is (real_SNo one_third H13R).
L173998
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L174000
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L174000
An exact proof term for the current goal is H0le_tmp.
L174001
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L174003
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L174003
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) (minus_SNo one_third)).
L174005
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun f1s x) (minus_SNo one_third)) Hm23le0 H0le).
L174007
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L174010
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) 1 (minus_SNo one_third) Hf1sRx real_1 Hm13R HhiI3).
L174011
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L174013
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L174013
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L174014
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) (minus_SNo one_third)) two_thirds.
L174016
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L174019
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) (minus_SNo one_third)) Hm23R H23R Hr1Rx Hlow Hup).
L174023
Assume HxnotC1: ¬ (x ∈ C1).
L174023
We prove the intermediate claim HxX: x ∈ X.
L174025
An exact proof term for the current goal is (HAsubX x HxA).
L174025
We prove the intermediate claim HnotI1: ¬ (apply_fun f1s x ∈ I1).
L174027
Assume Hf1sI1': apply_fun f1s x ∈ I1.
L174027
We prove the intermediate claim Hf1sI1I: apply_fun f1s x ∈ I1 ∩ I.
L174029
An exact proof term for the current goal is (binintersectI I1 I (apply_fun f1s x) Hf1sI1' Hf1sIx).
L174029
We prove the intermediate claim HxB1': x ∈ B1.
L174031
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f1s x0 ∈ I1 ∩ I) x HxA Hf1sI1I).
L174031
Apply FalseE to the current goal.
L174032
An exact proof term for the current goal is (HxnotB1 HxB1').
L174033
We prove the intermediate claim HnotI3: ¬ (apply_fun f1s x ∈ I3).
L174035
Assume Hf1sI3': apply_fun f1s x ∈ I3.
L174035
We prove the intermediate claim Hf1sI3I: apply_fun f1s x ∈ I3 ∩ I.
L174037
An exact proof term for the current goal is (binintersectI I3 I (apply_fun f1s x) Hf1sI3' Hf1sIx).
L174037
We prove the intermediate claim HxC1': x ∈ C1.
L174039
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun f1s x0 ∈ I3 ∩ I) x HxA Hf1sI3I).
L174039
Apply FalseE to the current goal.
L174040
An exact proof term for the current goal is (HxnotC1 HxC1').
L174041
We prove the intermediate claim Hf1sRx: apply_fun f1s x ∈ R.
L174043
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun f1s x) Hf1sIx).
L174043
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L174045
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174045
We prove the intermediate claim H13R: one_third ∈ R.
L174047
An exact proof term for the current goal is one_third_in_R.
L174047
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174049
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174049
We prove the intermediate claim Hf1s_boundsI: Rle (minus_SNo 1) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) 1.
L174051
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun f1s x) Hm1R real_1 Hf1sIx).
L174051
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun f1s x) (minus_SNo 1)).
L174053
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun f1s x) (andEL (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_boundsI)).
L174054
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun f1s x)).
L174056
An exact proof term for the current goal is (RleE_nlt (apply_fun f1s x) 1 (andER (Rle (minus_SNo 1) (apply_fun f1s x)) (Rle (apply_fun f1s x) 1) Hf1s_boundsI)).
L174057
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun f1s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun f1s x).
L174059
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun f1s x) Hm1R Hm13R Hf1sRx HnotI1).
L174060
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun f1s x).
L174062
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun f1s x))) to the current goal.
L174063
Assume Hbad: Rlt (apply_fun f1s x) (minus_SNo 1).
L174063
Apply FalseE to the current goal.
L174064
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L174066
Assume Hok: Rlt (minus_SNo one_third) (apply_fun f1s x).
L174066
An exact proof term for the current goal is Hok.
L174067
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun f1s x) (minus_SNo one_third)).
L174069
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun f1s x) Hm13lt_fx).
L174069
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun f1s x) one_third ∨ Rlt 1 (apply_fun f1s x).
L174071
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun f1s x) H13R real_1 Hf1sRx HnotI3).
L174072
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun f1s x) one_third.
L174074
Apply (HnotI3_cases (Rlt (apply_fun f1s x) one_third)) to the current goal.
L174075
Assume Hok: Rlt (apply_fun f1s x) one_third.
L174075
An exact proof term for the current goal is Hok.
L174077
Assume Hbad: Rlt 1 (apply_fun f1s x).
L174077
Apply FalseE to the current goal.
L174078
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L174079
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun f1s x)).
L174081
An exact proof term for the current goal is (not_Rlt_sym (apply_fun f1s x) one_third Hfx_lt_13).
L174081
We prove the intermediate claim Hf1sI0: apply_fun f1s x ∈ I0.
L174083
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L174085
We prove the intermediate claim HxSep: apply_fun f1s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L174087
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun f1s x) Hf1sRx (andI (¬ (Rlt (apply_fun f1s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun f1s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L174091
rewrite the current goal using HI0_def (from left to right).
L174092
An exact proof term for the current goal is HxSep.
L174093
We prove the intermediate claim Hu1funI0: function_on u1 X I0.
L174095
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u1 Hu1contI0).
L174095
We prove the intermediate claim Hu1xI0: apply_fun u1 x ∈ I0.
L174097
An exact proof term for the current goal is (Hu1funI0 x HxX).
L174097
We prove the intermediate claim Hu1xR: apply_fun u1 x ∈ R.
L174099
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u1 x) Hu1xI0).
L174099
We prove the intermediate claim Hm_u1x_R: minus_SNo (apply_fun u1 x) ∈ R.
L174101
An exact proof term for the current goal is (real_minus_SNo (apply_fun u1 x) Hu1xR).
L174101
rewrite the current goal using (Hr1_apply x HxA) (from left to right).
L174102
We prove the intermediate claim Hr1xR: add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) ∈ R.
L174104
An exact proof term for the current goal is (real_add_SNo (apply_fun f1s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) (minus_SNo (apply_fun u1 x)) Hm_u1x_R).
L174106
We prove the intermediate claim H23R: two_thirds ∈ R.
L174108
An exact proof term for the current goal is two_thirds_in_R.
L174108
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174110
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174110
We prove the intermediate claim Hf1s_bounds0: Rle (minus_SNo one_third) (apply_fun f1s x) ∧ Rle (apply_fun f1s x) one_third.
L174112
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun f1s x) Hm13R H13R Hf1sI0).
L174112
We prove the intermediate claim Hu1_bounds0: Rle (minus_SNo one_third) (apply_fun u1 x) ∧ Rle (apply_fun u1 x) one_third.
L174114
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u1 x) Hm13R H13R Hu1xI0).
L174114
We prove the intermediate claim Hm13_le_f1s: Rle (minus_SNo one_third) (apply_fun f1s x).
L174116
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun f1s x)) (Rle (apply_fun f1s x) one_third) Hf1s_bounds0).
L174116
We prove the intermediate claim Hf1s_le_13: Rle (apply_fun f1s x) one_third.
L174118
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun f1s x)) (Rle (apply_fun f1s x) one_third) Hf1s_bounds0).
L174118
We prove the intermediate claim Hm13_le_u1x: Rle (minus_SNo one_third) (apply_fun u1 x).
L174120
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u1 x)) (Rle (apply_fun u1 x) one_third) Hu1_bounds0).
L174120
We prove the intermediate claim Hu1x_le_13: Rle (apply_fun u1 x) one_third.
L174122
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u1 x)) (Rle (apply_fun u1 x) one_third) Hu1_bounds0).
L174122
We prove the intermediate claim Hm13_le_mu1: Rle (minus_SNo one_third) (minus_SNo (apply_fun u1 x)).
L174124
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u1 x) one_third Hu1x_le_13).
L174124
We prove the intermediate claim Hmu1_le_13: Rle (minus_SNo (apply_fun u1 x)) one_third.
L174126
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u1 x)) (minus_SNo (minus_SNo one_third)).
L174127
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u1 x) Hm13_le_u1x).
L174127
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L174128
An exact proof term for the current goal is Htmp.
L174129
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))).
L174132
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u1 x)) Hm13R Hm13R Hm_u1x_R Hm13_le_mu1).
L174133
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L174136
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) Hm_u1x_R Hm13_le_f1s).
L174140
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L174143
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) Hlow1 Hlow2).
L174146
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))).
L174148
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L174148
An exact proof term for the current goal is Hlow_tmp.
L174149
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) one_third).
L174152
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun f1s x) (minus_SNo (apply_fun u1 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) Hm_u1x_R H13R Hmu1_le_13).
L174154
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun f1s x) one_third) (add_SNo one_third one_third).
L174156
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun f1s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun f1s x) Hf1sI0) H13R H13R Hf1s_le_13).
L174158
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo one_third one_third).
L174161
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) (add_SNo (apply_fun f1s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L174164
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L174166
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) two_thirds.
L174168
rewrite the current goal using Hdef23 (from left to right) at position 1.
L174168
An exact proof term for the current goal is Hup_tmp.
L174169
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun f1s x) (minus_SNo (apply_fun u1 x))) Hm23R H23R Hr1xR Hlow Hup).
L174172
Set r1s to be the term compose_fun A r1 (div_const_fun den).
L174175
We prove the intermediate claim Hr1s_cont: continuous_map A Ta I Ti r1s.
(*** second-step scaled residual and second correction u2 ***)
L174177
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
(*** prove continuity into R, then restrict the range to I ***)
L174179
An exact proof term for the current goal is R_standard_topology_is_topology.
L174179
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L174181
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L174181
We prove the intermediate claim Hr1s_contR: continuous_map A Ta R R_standard_topology r1s.
L174183
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r1 (div_const_fun den) Hr1_cont Hdivcont).
L174184
We prove the intermediate claim Hr1s_I: ∀x : set, x ∈ A → apply_fun r1s x ∈ I.
L174186
Let x be given.
L174186
Assume HxA: x ∈ A.
L174186
We prove the intermediate claim Hr1xI2: apply_fun r1 x ∈ I2.
L174188
An exact proof term for the current goal is (Hr1_range x HxA).
L174188
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L174190
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174190
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L174192
An exact proof term for the current goal is (real_minus_SNo den H23R).
L174192
We prove the intermediate claim Hr1xR: apply_fun r1 x ∈ R.
L174194
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r1 x) Hr1xI2).
L174194
We prove the intermediate claim Hr1xS: SNo (apply_fun r1 x).
L174196
An exact proof term for the current goal is (real_SNo (apply_fun r1 x) Hr1xR).
L174196
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r1 x) ∧ Rle (apply_fun r1 x) den.
L174198
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r1 x) HmdenR H23R Hr1xI2).
L174198
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r1 x).
L174200
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r1 x)) (Rle (apply_fun r1 x) den) Hbounds).
L174202
We prove the intermediate claim Hhi: Rle (apply_fun r1 x) den.
L174204
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r1 x)) (Rle (apply_fun r1 x) den) Hbounds).
L174206
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r1 x)).
L174208
An exact proof term for the current goal is (RleE_nlt (apply_fun r1 x) den Hhi).
L174208
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r1 x) (minus_SNo den)).
L174210
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r1 x) Hlo).
L174210
We prove the intermediate claim HyEq: apply_fun r1s x = div_SNo (apply_fun r1 x) den.
L174212
rewrite the current goal using (compose_fun_apply A r1 (div_const_fun den) x HxA) (from left to right).
L174212
rewrite the current goal using (div_const_fun_apply den (apply_fun r1 x) H23R Hr1xR) (from left to right).
Use reflexivity.
L174214
We prove the intermediate claim HyR: apply_fun r1s x ∈ R.
L174216
rewrite the current goal using HyEq (from left to right).
L174216
An exact proof term for the current goal is (real_div_SNo (apply_fun r1 x) Hr1xR den H23R).
L174217
We prove the intermediate claim HyS: SNo (apply_fun r1s x).
L174219
An exact proof term for the current goal is (real_SNo (apply_fun r1s x) HyR).
L174219
We prove the intermediate claim Hy_le_1: Rle (apply_fun r1s x) 1.
L174221
Apply (RleI (apply_fun r1s x) 1 HyR real_1) to the current goal.
L174221
We will prove ¬ (Rlt 1 (apply_fun r1s x)).
L174222
Assume H1lt: Rlt 1 (apply_fun r1s x).
L174223
We prove the intermediate claim H1lty: 1 < apply_fun r1s x.
L174225
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r1s x) H1lt).
L174225
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r1s x) den.
L174227
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r1s x) den SNo_1 HyS H23S H23pos H1lty).
L174227
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r1s x) den.
L174229
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L174229
An exact proof term for the current goal is HmulLt.
L174230
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r1s x) den = apply_fun r1 x.
L174232
rewrite the current goal using HyEq (from left to right) at position 1.
L174232
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r1 x) den Hr1xS H23S H23ne0).
L174233
We prove the intermediate claim Hden_lt_r1x: den < apply_fun r1 x.
L174235
rewrite the current goal using HmulEq (from right to left).
L174235
An exact proof term for the current goal is HmulLt'.
L174236
We prove the intermediate claim Hbad: Rlt den (apply_fun r1 x).
L174238
An exact proof term for the current goal is (RltI den (apply_fun r1 x) H23R Hr1xR Hden_lt_r1x).
L174238
An exact proof term for the current goal is (Hnlt_hi Hbad).
L174239
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r1s x).
L174241
Apply (RleI (minus_SNo 1) (apply_fun r1s x) Hm1R HyR) to the current goal.
L174241
We will prove ¬ (Rlt (apply_fun r1s x) (minus_SNo 1)).
L174242
Assume Hylt: Rlt (apply_fun r1s x) (minus_SNo 1).
L174243
We prove the intermediate claim Hylts: apply_fun r1s x < minus_SNo 1.
L174245
An exact proof term for the current goal is (RltE_lt (apply_fun r1s x) (minus_SNo 1) Hylt).
L174245
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r1s x) den < mul_SNo (minus_SNo 1) den.
L174247
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r1s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L174248
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r1s x) den = apply_fun r1 x.
L174250
rewrite the current goal using HyEq (from left to right) at position 1.
L174250
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r1 x) den Hr1xS H23S H23ne0).
L174251
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L174253
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L174253
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L174255
We prove the intermediate claim Hr1x_lt_mden: apply_fun r1 x < minus_SNo den.
L174257
rewrite the current goal using HmulEq (from right to left).
L174257
rewrite the current goal using HrhsEq (from right to left).
L174258
An exact proof term for the current goal is HmulLt.
L174259
We prove the intermediate claim Hbad: Rlt (apply_fun r1 x) (minus_SNo den).
L174261
An exact proof term for the current goal is (RltI (apply_fun r1 x) (minus_SNo den) Hr1xR HmdenR Hr1x_lt_mden).
L174261
An exact proof term for the current goal is (Hnlt_lo Hbad).
L174262
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r1s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L174263
We will prove continuous_map A Ta I Ti r1s.
L174264
rewrite the current goal using HTiEq (from left to right).
L174265
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r1s I Hr1s_contR HIcR Hr1s_I).
L174266
We prove the intermediate claim Hex_u2: ∃u2 : set, continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third).
L174274
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r1s Hnorm HA Hr1s_cont).
L174274
Apply Hex_u2 to the current goal.
L174275
Let u2 be given.
L174276
Assume Hu2.
L174276
We prove the intermediate claim Hu2AB: continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third).
L174281
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third) Hu2).
L174287
We prove the intermediate claim Hu2contI0: continuous_map X Tx I0 T0 u2.
L174289
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u2) (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) Hu2AB).
L174293
We prove the intermediate claim Hu2contR: continuous_map X Tx R R_standard_topology u2.
L174295
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L174296
We prove the intermediate claim HI0subR: I0 ⊆ R.
L174298
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L174298
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u2 Hu2contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L174306
Set den2 to be the term mul_SNo den den.
L174309
We prove the intermediate claim Hden2R: den2 ∈ R.
(*** a second partial sum g2 = g1 + (2/3)^2 u2 is continuous ***)
L174311
An exact proof term for the current goal is (real_mul_SNo den HdenR den HdenR).
L174311
We prove the intermediate claim Hden2pos: 0 < den2.
L174313
An exact proof term for the current goal is (mul_SNo_pos_pos den den H23S H23S HdenPos HdenPos).
L174313
Set u2s to be the term compose_fun X u2 (mul_const_fun den2).
L174314
We prove the intermediate claim Hu2s_cont: continuous_map X Tx R R_standard_topology u2s.
L174316
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u2 den2 HTx Hu2contR Hden2R Hden2pos).
L174316
Set g2 to be the term compose_fun X (pair_map X g1 u2s) add_fun_R.
L174317
We prove the intermediate claim Hg2cont: continuous_map X Tx R R_standard_topology g2.
L174319
An exact proof term for the current goal is (add_two_continuous_R X Tx g1 u2s HTx Hg1cont Hu2s_cont).
L174319
We prove the intermediate claim Hu2contA: continuous_map A Ta R R_standard_topology u2.
(*** third correction step scaffold: residual r2 on A, scaling r2s, and u3 ***)
L174323
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u2 A HTx HAsubX Hu2contR).
L174323
Set u2neg to be the term compose_fun A u2 neg_fun.
L174324
We prove the intermediate claim Hu2neg_cont: continuous_map A Ta R R_standard_topology u2neg.
L174326
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u2 neg_fun Hu2contA Hnegcont).
L174327
We prove the intermediate claim Hr1s_contR: continuous_map A Ta R R_standard_topology r1s.
L174329
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r1s Hr1s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L174330
Set r2 to be the term compose_fun A (pair_map A r1s u2neg) add_fun_R.
L174331
We prove the intermediate claim Hr2_cont: continuous_map A Ta R R_standard_topology r2.
L174333
An exact proof term for the current goal is (add_two_continuous_R A Ta r1s u2neg HTa Hr1s_contR Hu2neg_cont).
L174333
We prove the intermediate claim Hr2_apply: ∀x : set, x ∈ A → apply_fun r2 x = add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x)).
L174336
Let x be given.
L174336
Assume HxA: x ∈ A.
L174336
We prove the intermediate claim Hpimg: apply_fun (pair_map A r1s u2neg) x ∈ setprod R R.
L174338
rewrite the current goal using (pair_map_apply A R R r1s u2neg x HxA) (from left to right).
L174338
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L174340
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology r1s Hr1s_contR x HxA).
L174340
We prove the intermediate claim Hu2negRx: apply_fun u2neg x ∈ R.
L174342
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u2neg Hu2neg_cont x HxA).
L174342
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r1s x) (apply_fun u2neg x) Hr1sRx Hu2negRx).
L174343
rewrite the current goal using (compose_fun_apply A (pair_map A r1s u2neg) add_fun_R x HxA) (from left to right).
L174344
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r1s u2neg) x) Hpimg) (from left to right) at position 1.
L174345
rewrite the current goal using (pair_map_apply A R R r1s u2neg x HxA) (from left to right).
L174346
rewrite the current goal using (tuple_2_0_eq (apply_fun r1s x) (apply_fun u2neg x)) (from left to right).
L174347
rewrite the current goal using (tuple_2_1_eq (apply_fun r1s x) (apply_fun u2neg x)) (from left to right).
L174348
rewrite the current goal using (compose_fun_apply A u2 neg_fun x HxA) (from left to right) at position 1.
L174349
We prove the intermediate claim Hu2Rx: apply_fun u2 x ∈ R.
L174351
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u2 Hu2contA x HxA).
L174351
rewrite the current goal using (neg_fun_apply (apply_fun u2 x) Hu2Rx) (from left to right) at position 1.
Use reflexivity.
L174353
We prove the intermediate claim Hr2_range: ∀x : set, x ∈ A → apply_fun r2 x ∈ I2.
L174355
We prove the intermediate claim Hu2_on_B2: ∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third.
(*** show the next residual stays within [-2/3,2/3] on A ***)
L174359
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u2) (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third) Hu2AB).
L174363
We prove the intermediate claim Hu2_on_C2: ∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third.
L174367
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u2 ∧ (∀x : set, x ∈ preimage_of A r1s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u2 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r1s ((closed_interval one_third 1) ∩ I) → apply_fun u2 x = one_third) Hu2).
L174373
Let x be given.
L174374
Assume HxA: x ∈ A.
L174374
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L174375
Set I3 to be the term closed_interval one_third 1.
L174376
Set B2 to be the term preimage_of A r1s (I1 ∩ I).
L174377
Set C2 to be the term preimage_of A r1s (I3 ∩ I).
L174378
We prove the intermediate claim Hr1sIx: apply_fun r1s x ∈ I.
L174380
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r1s Hr1s_cont x HxA).
L174380
We prove the intermediate claim HB2_cases: x ∈ B2 ∨ ¬ (x ∈ B2).
L174382
An exact proof term for the current goal is (xm (x ∈ B2)).
L174382
Apply (HB2_cases (apply_fun r2 x ∈ I2)) to the current goal.
L174384
Assume HxB2: x ∈ B2.
L174384
We prove the intermediate claim Hu2eq: apply_fun u2 x = minus_SNo one_third.
L174386
An exact proof term for the current goal is (Hu2_on_B2 x HxB2).
L174386
We prove the intermediate claim Hr2eq: apply_fun r2 x = add_SNo (apply_fun r1s x) one_third.
L174388
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L174388
rewrite the current goal using Hu2eq (from left to right) at position 1.
L174389
We prove the intermediate claim H13R: one_third ∈ R.
L174391
An exact proof term for the current goal is one_third_in_R.
L174391
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L174393
rewrite the current goal using Hr2eq (from left to right).
L174394
We prove the intermediate claim Hr1sI1I: apply_fun r1s x ∈ I1 ∩ I.
L174396
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r1s x0 ∈ I1 ∩ I) x HxB2).
L174396
We prove the intermediate claim Hr1sI1: apply_fun r1s x ∈ I1.
L174398
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r1s x) Hr1sI1I).
L174398
We prove the intermediate claim H13R: one_third ∈ R.
L174400
An exact proof term for the current goal is one_third_in_R.
L174400
We prove the intermediate claim H23R: two_thirds ∈ R.
L174402
An exact proof term for the current goal is two_thirds_in_R.
L174402
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174404
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174404
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L174406
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174406
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174408
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174408
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L174410
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hr1sI1).
L174410
We prove the intermediate claim Hr1s_bounds: Rle (minus_SNo 1) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) (minus_SNo one_third).
L174412
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hm1R Hm13R Hr1sI1).
L174413
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r1s x).
L174415
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) (minus_SNo one_third)) Hr1s_bounds).
L174416
We prove the intermediate claim HhiI1: Rle (apply_fun r1s x) (minus_SNo one_third).
L174418
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) (minus_SNo one_third)) Hr1s_bounds).
L174419
We prove the intermediate claim Hr2Rx: add_SNo (apply_fun r1s x) one_third ∈ R.
L174421
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) Hr1sRx one_third H13R).
L174421
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r1s x) one_third).
L174423
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r1s x) one_third Hm1R Hr1sRx H13R Hm1le).
L174423
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) one_third).
L174425
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L174425
An exact proof term for the current goal is Hlow_tmp.
L174426
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r1s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L174428
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) (minus_SNo one_third) one_third Hr1sRx Hm13R H13R HhiI1).
L174428
We prove the intermediate claim H13S: SNo one_third.
L174430
An exact proof term for the current goal is (real_SNo one_third H13R).
L174430
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r1s x) one_third) 0.
L174432
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L174432
An exact proof term for the current goal is Hup0_tmp.
L174433
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L174435
An exact proof term for the current goal is Rle_0_two_thirds.
L174435
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) one_third) two_thirds.
L174437
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) one_third) 0 two_thirds Hup0 H0le23).
L174437
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) one_third) Hm23R H23R Hr2Rx Hlow Hup).
L174441
Assume HxnotB2: ¬ (x ∈ B2).
L174441
We prove the intermediate claim HC2_cases: x ∈ C2 ∨ ¬ (x ∈ C2).
L174443
An exact proof term for the current goal is (xm (x ∈ C2)).
L174443
Apply (HC2_cases (apply_fun r2 x ∈ I2)) to the current goal.
L174445
Assume HxC2: x ∈ C2.
L174445
We prove the intermediate claim Hu2eq: apply_fun u2 x = one_third.
L174447
An exact proof term for the current goal is (Hu2_on_C2 x HxC2).
L174447
We prove the intermediate claim Hr2eq: apply_fun r2 x = add_SNo (apply_fun r1s x) (minus_SNo one_third).
L174449
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L174449
rewrite the current goal using Hu2eq (from left to right) at position 1.
Use reflexivity.
L174451
rewrite the current goal using Hr2eq (from left to right).
L174452
We prove the intermediate claim Hr1sI3I: apply_fun r1s x ∈ I3 ∩ I.
L174454
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r1s x0 ∈ I3 ∩ I) x HxC2).
L174454
We prove the intermediate claim Hr1sI3: apply_fun r1s x ∈ I3.
L174456
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r1s x) Hr1sI3I).
L174456
We prove the intermediate claim H13R: one_third ∈ R.
L174458
An exact proof term for the current goal is one_third_in_R.
L174458
We prove the intermediate claim H23R: two_thirds ∈ R.
L174460
An exact proof term for the current goal is two_thirds_in_R.
L174460
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174462
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174462
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174464
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174464
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L174466
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r1s x) Hr1sI3).
L174466
We prove the intermediate claim Hr1s_bounds: Rle one_third (apply_fun r1s x) ∧ Rle (apply_fun r1s x) 1.
L174468
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r1s x) H13R real_1 Hr1sI3).
L174468
We prove the intermediate claim HloI3: Rle one_third (apply_fun r1s x).
L174470
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_bounds).
L174470
We prove the intermediate claim HhiI3: Rle (apply_fun r1s x) 1.
L174472
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_bounds).
L174472
We prove the intermediate claim Hr2Rx: add_SNo (apply_fun r1s x) (minus_SNo one_third) ∈ R.
L174474
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) Hr1sRx (minus_SNo one_third) Hm13R).
L174474
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L174477
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r1s x) (minus_SNo one_third) H13R Hr1sRx Hm13R HloI3).
L174478
We prove the intermediate claim H13S: SNo one_third.
L174480
An exact proof term for the current goal is (real_SNo one_third H13R).
L174480
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L174482
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L174482
An exact proof term for the current goal is H0le_tmp.
L174483
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L174485
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L174485
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) (minus_SNo one_third)).
L174487
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r1s x) (minus_SNo one_third)) Hm23le0 H0le).
L174489
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L174492
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) 1 (minus_SNo one_third) Hr1sRx real_1 Hm13R HhiI3).
L174493
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L174495
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L174495
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L174496
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) (minus_SNo one_third)) two_thirds.
L174498
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L174501
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) (minus_SNo one_third)) Hm23R H23R Hr2Rx Hlow Hup).
L174505
Assume HxnotC2: ¬ (x ∈ C2).
L174505
We prove the intermediate claim HxX: x ∈ X.
L174507
An exact proof term for the current goal is (HAsubX x HxA).
L174507
We prove the intermediate claim HnotI1: ¬ (apply_fun r1s x ∈ I1).
L174509
Assume Hr1sI1': apply_fun r1s x ∈ I1.
L174509
We prove the intermediate claim Hr1sI1I: apply_fun r1s x ∈ I1 ∩ I.
L174511
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r1s x) Hr1sI1' Hr1sIx).
L174511
We prove the intermediate claim HxB2': x ∈ B2.
L174513
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r1s x0 ∈ I1 ∩ I) x HxA Hr1sI1I).
L174513
Apply FalseE to the current goal.
L174514
An exact proof term for the current goal is (HxnotB2 HxB2').
L174515
We prove the intermediate claim HnotI3: ¬ (apply_fun r1s x ∈ I3).
L174517
Assume Hr1sI3': apply_fun r1s x ∈ I3.
L174517
We prove the intermediate claim Hr1sI3I: apply_fun r1s x ∈ I3 ∩ I.
L174519
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r1s x) Hr1sI3' Hr1sIx).
L174519
We prove the intermediate claim HxC2': x ∈ C2.
L174521
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r1s x0 ∈ I3 ∩ I) x HxA Hr1sI3I).
L174521
Apply FalseE to the current goal.
L174522
An exact proof term for the current goal is (HxnotC2 HxC2').
L174523
We prove the intermediate claim Hr1sRx: apply_fun r1s x ∈ R.
L174525
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r1s x) Hr1sIx).
L174525
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L174527
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174527
We prove the intermediate claim H13R: one_third ∈ R.
L174529
An exact proof term for the current goal is one_third_in_R.
L174529
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174531
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174531
We prove the intermediate claim Hr1s_boundsI: Rle (minus_SNo 1) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) 1.
L174533
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r1s x) Hm1R real_1 Hr1sIx).
L174533
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r1s x) (minus_SNo 1)).
L174535
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r1s x) (andEL (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_boundsI)).
L174536
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r1s x)).
L174538
An exact proof term for the current goal is (RleE_nlt (apply_fun r1s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r1s x)) (Rle (apply_fun r1s x) 1) Hr1s_boundsI)).
L174539
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r1s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r1s x).
L174541
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r1s x) Hm1R Hm13R Hr1sRx HnotI1).
L174542
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r1s x).
L174544
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r1s x))) to the current goal.
L174545
Assume Hbad: Rlt (apply_fun r1s x) (minus_SNo 1).
L174545
Apply FalseE to the current goal.
L174546
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L174548
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r1s x).
L174548
An exact proof term for the current goal is Hok.
L174549
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r1s x) (minus_SNo one_third)).
L174551
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r1s x) Hm13lt_fx).
L174551
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r1s x) one_third ∨ Rlt 1 (apply_fun r1s x).
L174553
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r1s x) H13R real_1 Hr1sRx HnotI3).
L174554
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r1s x) one_third.
L174556
Apply (HnotI3_cases (Rlt (apply_fun r1s x) one_third)) to the current goal.
L174557
Assume Hok: Rlt (apply_fun r1s x) one_third.
L174557
An exact proof term for the current goal is Hok.
L174559
Assume Hbad: Rlt 1 (apply_fun r1s x).
L174559
Apply FalseE to the current goal.
L174560
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L174561
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r1s x)).
L174563
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r1s x) one_third Hfx_lt_13).
L174563
We prove the intermediate claim Hr1sI0: apply_fun r1s x ∈ I0.
L174565
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L174567
We prove the intermediate claim HxSep: apply_fun r1s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L174569
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r1s x) Hr1sRx (andI (¬ (Rlt (apply_fun r1s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r1s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L174573
rewrite the current goal using HI0_def (from left to right).
L174574
An exact proof term for the current goal is HxSep.
L174575
We prove the intermediate claim Hu2funI0: function_on u2 X I0.
L174577
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u2 Hu2contI0).
L174577
We prove the intermediate claim Hu2xI0: apply_fun u2 x ∈ I0.
L174579
An exact proof term for the current goal is (Hu2funI0 x HxX).
L174579
We prove the intermediate claim Hu2xR: apply_fun u2 x ∈ R.
L174581
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u2 x) Hu2xI0).
L174581
We prove the intermediate claim Hm_u2x_R: minus_SNo (apply_fun u2 x) ∈ R.
L174583
An exact proof term for the current goal is (real_minus_SNo (apply_fun u2 x) Hu2xR).
L174583
rewrite the current goal using (Hr2_apply x HxA) (from left to right).
L174584
We prove the intermediate claim Hr2xR: add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) ∈ R.
L174586
An exact proof term for the current goal is (real_add_SNo (apply_fun r1s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) (minus_SNo (apply_fun u2 x)) Hm_u2x_R).
L174588
We prove the intermediate claim H23R: two_thirds ∈ R.
L174590
An exact proof term for the current goal is two_thirds_in_R.
L174590
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174592
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174592
We prove the intermediate claim Hr1s_bounds0: Rle (minus_SNo one_third) (apply_fun r1s x) ∧ Rle (apply_fun r1s x) one_third.
L174594
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r1s x) Hm13R H13R Hr1sI0).
L174594
We prove the intermediate claim Hu2_bounds0: Rle (minus_SNo one_third) (apply_fun u2 x) ∧ Rle (apply_fun u2 x) one_third.
L174596
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u2 x) Hm13R H13R Hu2xI0).
L174596
We prove the intermediate claim Hm13_le_r1s: Rle (minus_SNo one_third) (apply_fun r1s x).
L174598
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r1s x)) (Rle (apply_fun r1s x) one_third) Hr1s_bounds0).
L174598
We prove the intermediate claim Hr1s_le_13: Rle (apply_fun r1s x) one_third.
L174600
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r1s x)) (Rle (apply_fun r1s x) one_third) Hr1s_bounds0).
L174600
We prove the intermediate claim Hm13_le_u2x: Rle (minus_SNo one_third) (apply_fun u2 x).
L174602
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u2 x)) (Rle (apply_fun u2 x) one_third) Hu2_bounds0).
L174602
We prove the intermediate claim Hu2x_le_13: Rle (apply_fun u2 x) one_third.
L174604
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u2 x)) (Rle (apply_fun u2 x) one_third) Hu2_bounds0).
L174604
We prove the intermediate claim Hm13_le_mu2: Rle (minus_SNo one_third) (minus_SNo (apply_fun u2 x)).
L174606
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u2 x) one_third Hu2x_le_13).
L174606
We prove the intermediate claim Hmu2_le_13: Rle (minus_SNo (apply_fun u2 x)) one_third.
L174608
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u2 x)) (minus_SNo (minus_SNo one_third)).
L174609
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u2 x) Hm13_le_u2x).
L174609
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L174610
An exact proof term for the current goal is Htmp.
L174611
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))).
L174614
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u2 x)) Hm13R Hm13R Hm_u2x_R Hm13_le_mu2).
L174615
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L174618
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) Hm_u2x_R Hm13_le_r1s).
L174622
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L174625
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) Hlow1 Hlow2).
L174628
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))).
L174630
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L174630
An exact proof term for the current goal is Hlow_tmp.
L174631
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) one_third).
L174634
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r1s x) (minus_SNo (apply_fun u2 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) Hm_u2x_R H13R Hmu2_le_13).
L174636
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r1s x) one_third) (add_SNo one_third one_third).
L174639
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r1s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r1s x) Hr1sI0) H13R H13R Hr1s_le_13).
L174641
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo one_third one_third).
L174644
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) (add_SNo (apply_fun r1s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L174647
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) two_thirds.
L174649
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L174650
rewrite the current goal using Hdef23 (from left to right).
L174651
An exact proof term for the current goal is Hup_tmp.
L174652
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r1s x) (minus_SNo (apply_fun u2 x))) Hm23R H23R Hr2xR Hlow Hup).
L174655
Set r2s to be the term compose_fun A r2 (div_const_fun den).
L174656
We prove the intermediate claim Hr2s_cont: continuous_map A Ta I Ti r2s.
L174658
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L174659
An exact proof term for the current goal is R_standard_topology_is_topology.
L174659
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L174661
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L174661
We prove the intermediate claim Hr2s_contR: continuous_map A Ta R R_standard_topology r2s.
L174663
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r2 (div_const_fun den) Hr2_cont Hdivcont).
L174664
We prove the intermediate claim Hr2s_I: ∀x : set, x ∈ A → apply_fun r2s x ∈ I.
L174666
Let x be given.
L174666
Assume HxA: x ∈ A.
L174666
We prove the intermediate claim Hr2xI2: apply_fun r2 x ∈ I2.
L174668
An exact proof term for the current goal is (Hr2_range x HxA).
L174668
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L174670
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174670
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L174672
An exact proof term for the current goal is (real_minus_SNo den H23R).
L174672
We prove the intermediate claim Hr2xR: apply_fun r2 x ∈ R.
L174674
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r2 x) Hr2xI2).
L174674
We prove the intermediate claim Hr2xS: SNo (apply_fun r2 x).
L174676
An exact proof term for the current goal is (real_SNo (apply_fun r2 x) Hr2xR).
L174676
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r2 x) ∧ Rle (apply_fun r2 x) den.
L174678
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r2 x) HmdenR H23R Hr2xI2).
L174678
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r2 x).
L174680
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r2 x)) (Rle (apply_fun r2 x) den) Hbounds).
L174682
We prove the intermediate claim Hhi: Rle (apply_fun r2 x) den.
L174684
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r2 x)) (Rle (apply_fun r2 x) den) Hbounds).
L174686
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r2 x)).
L174688
An exact proof term for the current goal is (RleE_nlt (apply_fun r2 x) den Hhi).
L174688
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r2 x) (minus_SNo den)).
L174690
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r2 x) Hlo).
L174690
We prove the intermediate claim HyEq: apply_fun r2s x = div_SNo (apply_fun r2 x) den.
L174692
rewrite the current goal using (compose_fun_apply A r2 (div_const_fun den) x HxA) (from left to right).
L174692
rewrite the current goal using (div_const_fun_apply den (apply_fun r2 x) H23R Hr2xR) (from left to right).
Use reflexivity.
L174694
We prove the intermediate claim HyR: apply_fun r2s x ∈ R.
L174696
rewrite the current goal using HyEq (from left to right).
L174696
An exact proof term for the current goal is (real_div_SNo (apply_fun r2 x) Hr2xR den H23R).
L174697
We prove the intermediate claim HyS: SNo (apply_fun r2s x).
L174699
An exact proof term for the current goal is (real_SNo (apply_fun r2s x) HyR).
L174699
We prove the intermediate claim Hy_le_1: Rle (apply_fun r2s x) 1.
L174701
Apply (RleI (apply_fun r2s x) 1 HyR real_1) to the current goal.
L174701
We will prove ¬ (Rlt 1 (apply_fun r2s x)).
L174702
Assume H1lt: Rlt 1 (apply_fun r2s x).
L174703
We prove the intermediate claim H1lty: 1 < apply_fun r2s x.
L174705
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r2s x) H1lt).
L174705
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r2s x) den.
L174707
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r2s x) den SNo_1 HyS H23S H23pos H1lty).
L174707
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r2s x) den.
L174709
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L174709
An exact proof term for the current goal is HmulLt.
L174710
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r2s x) den = apply_fun r2 x.
L174712
rewrite the current goal using HyEq (from left to right) at position 1.
L174712
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r2 x) den Hr2xS H23S H23ne0).
L174713
We prove the intermediate claim Hden_lt_r2x: den < apply_fun r2 x.
L174715
rewrite the current goal using HmulEq (from right to left).
L174715
An exact proof term for the current goal is HmulLt'.
L174716
We prove the intermediate claim Hbad: Rlt den (apply_fun r2 x).
L174718
An exact proof term for the current goal is (RltI den (apply_fun r2 x) H23R Hr2xR Hden_lt_r2x).
L174718
An exact proof term for the current goal is (Hnlt_hi Hbad).
L174719
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r2s x).
L174721
Apply (RleI (minus_SNo 1) (apply_fun r2s x) Hm1R HyR) to the current goal.
L174721
We will prove ¬ (Rlt (apply_fun r2s x) (minus_SNo 1)).
L174722
Assume Hylt: Rlt (apply_fun r2s x) (minus_SNo 1).
L174723
We prove the intermediate claim Hylts: apply_fun r2s x < minus_SNo 1.
L174725
An exact proof term for the current goal is (RltE_lt (apply_fun r2s x) (minus_SNo 1) Hylt).
L174725
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r2s x) den < mul_SNo (minus_SNo 1) den.
L174727
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r2s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L174728
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r2s x) den = apply_fun r2 x.
L174730
rewrite the current goal using HyEq (from left to right) at position 1.
L174730
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r2 x) den Hr2xS H23S H23ne0).
L174731
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L174733
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L174733
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L174735
We prove the intermediate claim Hr2x_lt_mden: apply_fun r2 x < minus_SNo den.
L174737
rewrite the current goal using HmulEq (from right to left).
L174737
rewrite the current goal using HrhsEq (from right to left).
L174738
An exact proof term for the current goal is HmulLt.
L174739
We prove the intermediate claim Hbad: Rlt (apply_fun r2 x) (minus_SNo den).
L174741
An exact proof term for the current goal is (RltI (apply_fun r2 x) (minus_SNo den) Hr2xR HmdenR Hr2x_lt_mden).
L174741
An exact proof term for the current goal is (Hnlt_lo Hbad).
L174742
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r2s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L174744
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r2s I Hr2s_contR HIcR Hr2s_I).
L174745
We prove the intermediate claim Hex_u3: ∃u3 : set, continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third).
L174753
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r2s Hnorm HA Hr2s_cont).
L174753
Apply Hex_u3 to the current goal.
L174754
Let u3 be given.
L174755
Assume Hu3.
L174755
We prove the intermediate claim Hu3contI0: continuous_map X Tx I0 T0 u3.
L174757
We prove the intermediate claim Hu3AB: continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third).
L174761
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L174767
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3) (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) Hu3AB).
L174772
We prove the intermediate claim Hu3contR: continuous_map X Tx R R_standard_topology u3.
L174774
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L174775
We prove the intermediate claim HI0subR: I0 ⊆ R.
L174777
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L174777
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u3 Hu3contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L174785
Set den3 to be the term mul_SNo den2 den.
L174786
We prove the intermediate claim Hden3R: den3 ∈ R.
L174788
An exact proof term for the current goal is (real_mul_SNo den2 Hden2R den HdenR).
L174788
We prove the intermediate claim Hden3pos: 0 < den3.
L174790
We prove the intermediate claim Hden2S: SNo den2.
L174791
An exact proof term for the current goal is (real_SNo den2 Hden2R).
L174791
An exact proof term for the current goal is (mul_SNo_pos_pos den2 den Hden2S H23S Hden2pos HdenPos).
L174792
Set u3s to be the term compose_fun X u3 (mul_const_fun den3).
L174793
We prove the intermediate claim Hu3s_cont: continuous_map X Tx R R_standard_topology u3s.
L174795
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u3 den3 HTx Hu3contR Hden3R Hden3pos).
L174795
Set g3 to be the term compose_fun X (pair_map X g2 u3s) add_fun_R.
L174796
We prove the intermediate claim Hg3cont: continuous_map X Tx R R_standard_topology g3.
L174798
An exact proof term for the current goal is (add_two_continuous_R X Tx g2 u3s HTx Hg2cont Hu3s_cont).
L174798
We prove the intermediate claim Hu3contA: continuous_map A Ta R R_standard_topology u3.
(*** fourth correction step scaffold: residual r3 on A, scaling r3s, and u4 ***)
L174802
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u3 A HTx HAsubX Hu3contR).
L174802
Set u3neg to be the term compose_fun A u3 neg_fun.
L174803
We prove the intermediate claim Hu3neg_cont: continuous_map A Ta R R_standard_topology u3neg.
L174805
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u3 neg_fun Hu3contA Hnegcont).
L174806
We prove the intermediate claim Hr2s_contR: continuous_map A Ta R R_standard_topology r2s.
L174808
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r2s Hr2s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L174809
Set r3 to be the term compose_fun A (pair_map A r2s u3neg) add_fun_R.
L174810
We prove the intermediate claim Hr3_cont: continuous_map A Ta R R_standard_topology r3.
L174812
An exact proof term for the current goal is (add_two_continuous_R A Ta r2s u3neg HTa Hr2s_contR Hu3neg_cont).
L174812
We prove the intermediate claim Hr3_apply: ∀x : set, x ∈ A → apply_fun r3 x = add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x)).
L174815
Let x be given.
L174815
Assume HxA: x ∈ A.
L174815
We prove the intermediate claim Hpimg: apply_fun (pair_map A r2s u3neg) x ∈ setprod R R.
L174817
rewrite the current goal using (pair_map_apply A R R r2s u3neg x HxA) (from left to right).
L174817
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L174819
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology r2s Hr2s_contR x HxA).
L174819
We prove the intermediate claim Hu3negRx: apply_fun u3neg x ∈ R.
L174821
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u3neg Hu3neg_cont x HxA).
L174821
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r2s x) (apply_fun u3neg x) Hr2sRx Hu3negRx).
L174822
rewrite the current goal using (compose_fun_apply A (pair_map A r2s u3neg) add_fun_R x HxA) (from left to right).
L174823
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r2s u3neg) x) Hpimg) (from left to right) at position 1.
L174824
rewrite the current goal using (pair_map_apply A R R r2s u3neg x HxA) (from left to right).
L174825
rewrite the current goal using (tuple_2_0_eq (apply_fun r2s x) (apply_fun u3neg x)) (from left to right).
L174826
rewrite the current goal using (tuple_2_1_eq (apply_fun r2s x) (apply_fun u3neg x)) (from left to right).
L174827
rewrite the current goal using (compose_fun_apply A u3 neg_fun x HxA) (from left to right) at position 1.
L174828
We prove the intermediate claim Hu3Rx: apply_fun u3 x ∈ R.
L174830
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u3 Hu3contA x HxA).
L174830
rewrite the current goal using (neg_fun_apply (apply_fun u3 x) Hu3Rx) (from left to right) at position 1.
Use reflexivity.
L174832
We prove the intermediate claim Hr3_range: ∀x : set, x ∈ A → apply_fun r3 x ∈ I2.
L174834
We prove the intermediate claim Hu3AB: continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L174839
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L174845
We prove the intermediate claim Hu3_on_B3: ∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third.
L174849
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u3) (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third) Hu3AB).
L174853
We prove the intermediate claim Hu3_on_C3: ∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third.
L174857
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u3 ∧ (∀x : set, x ∈ preimage_of A r2s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u3 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r2s ((closed_interval one_third 1) ∩ I) → apply_fun u3 x = one_third) Hu3).
L174863
Let x be given.
L174864
Assume HxA: x ∈ A.
L174864
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L174865
Set I3 to be the term closed_interval one_third 1.
L174866
Set B3 to be the term preimage_of A r2s (I1 ∩ I).
L174867
Set C3 to be the term preimage_of A r2s (I3 ∩ I).
L174868
We prove the intermediate claim Hr2sIx: apply_fun r2s x ∈ I.
L174870
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r2s Hr2s_cont x HxA).
L174870
We prove the intermediate claim HB3_cases: x ∈ B3 ∨ ¬ (x ∈ B3).
L174872
An exact proof term for the current goal is (xm (x ∈ B3)).
L174872
Apply (HB3_cases (apply_fun r3 x ∈ I2)) to the current goal.
L174874
Assume HxB3: x ∈ B3.
L174874
We prove the intermediate claim Hu3eq: apply_fun u3 x = minus_SNo one_third.
L174876
An exact proof term for the current goal is (Hu3_on_B3 x HxB3).
L174876
We prove the intermediate claim Hr3eq: apply_fun r3 x = add_SNo (apply_fun r2s x) one_third.
L174878
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L174878
rewrite the current goal using Hu3eq (from left to right) at position 1.
L174879
We prove the intermediate claim H13R: one_third ∈ R.
L174881
An exact proof term for the current goal is one_third_in_R.
L174881
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L174883
rewrite the current goal using Hr3eq (from left to right).
L174884
We prove the intermediate claim Hr2sI1I: apply_fun r2s x ∈ I1 ∩ I.
L174886
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r2s x0 ∈ I1 ∩ I) x HxB3).
L174886
We prove the intermediate claim Hr2sI1: apply_fun r2s x ∈ I1.
L174888
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r2s x) Hr2sI1I).
L174888
We prove the intermediate claim H13R: one_third ∈ R.
L174890
An exact proof term for the current goal is one_third_in_R.
L174890
We prove the intermediate claim H23R: two_thirds ∈ R.
L174892
An exact proof term for the current goal is two_thirds_in_R.
L174892
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174894
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174894
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L174896
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L174896
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174898
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174898
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L174900
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hr2sI1).
L174900
We prove the intermediate claim Hr2s_bounds: Rle (minus_SNo 1) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) (minus_SNo one_third).
L174902
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hm1R Hm13R Hr2sI1).
L174903
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r2s x).
L174905
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) (minus_SNo one_third)) Hr2s_bounds).
L174906
We prove the intermediate claim HhiI1: Rle (apply_fun r2s x) (minus_SNo one_third).
L174908
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) (minus_SNo one_third)) Hr2s_bounds).
L174909
We prove the intermediate claim Hr3Rx: add_SNo (apply_fun r2s x) one_third ∈ R.
L174911
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) Hr2sRx one_third H13R).
L174911
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r2s x) one_third).
L174913
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r2s x) one_third Hm1R Hr2sRx H13R Hm1le).
L174913
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) one_third).
L174915
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L174915
An exact proof term for the current goal is Hlow_tmp.
L174916
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r2s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L174918
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) (minus_SNo one_third) one_third Hr2sRx Hm13R H13R HhiI1).
L174918
We prove the intermediate claim H13S: SNo one_third.
L174920
An exact proof term for the current goal is (real_SNo one_third H13R).
L174920
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r2s x) one_third) 0.
L174922
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L174922
An exact proof term for the current goal is Hup0_tmp.
L174923
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L174925
An exact proof term for the current goal is Rle_0_two_thirds.
L174925
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) one_third) two_thirds.
L174927
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) one_third) 0 two_thirds Hup0 H0le23).
L174927
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) one_third) Hm23R H23R Hr3Rx Hlow Hup).
L174931
Assume HxnotB3: ¬ (x ∈ B3).
L174931
We prove the intermediate claim HC3_cases: x ∈ C3 ∨ ¬ (x ∈ C3).
L174933
An exact proof term for the current goal is (xm (x ∈ C3)).
L174933
Apply (HC3_cases (apply_fun r3 x ∈ I2)) to the current goal.
L174935
Assume HxC3: x ∈ C3.
L174935
We prove the intermediate claim Hu3eq: apply_fun u3 x = one_third.
L174937
An exact proof term for the current goal is (Hu3_on_C3 x HxC3).
L174937
We prove the intermediate claim Hr3eq: apply_fun r3 x = add_SNo (apply_fun r2s x) (minus_SNo one_third).
L174939
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L174939
rewrite the current goal using Hu3eq (from left to right) at position 1.
Use reflexivity.
L174941
rewrite the current goal using Hr3eq (from left to right).
L174942
We prove the intermediate claim Hr2sI3I: apply_fun r2s x ∈ I3 ∩ I.
L174944
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r2s x0 ∈ I3 ∩ I) x HxC3).
L174944
We prove the intermediate claim Hr2sI3: apply_fun r2s x ∈ I3.
L174946
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r2s x) Hr2sI3I).
L174946
We prove the intermediate claim H13R: one_third ∈ R.
L174948
An exact proof term for the current goal is one_third_in_R.
L174948
We prove the intermediate claim H23R: two_thirds ∈ R.
L174950
An exact proof term for the current goal is two_thirds_in_R.
L174950
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L174952
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L174952
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L174954
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L174954
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L174956
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r2s x) Hr2sI3).
L174956
We prove the intermediate claim Hr2s_bounds: Rle one_third (apply_fun r2s x) ∧ Rle (apply_fun r2s x) 1.
L174958
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r2s x) H13R real_1 Hr2sI3).
L174958
We prove the intermediate claim HloI3: Rle one_third (apply_fun r2s x).
L174960
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_bounds).
L174960
We prove the intermediate claim HhiI3: Rle (apply_fun r2s x) 1.
L174962
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_bounds).
L174962
We prove the intermediate claim Hr3Rx: add_SNo (apply_fun r2s x) (minus_SNo one_third) ∈ R.
L174964
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) Hr2sRx (minus_SNo one_third) Hm13R).
L174964
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L174967
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r2s x) (minus_SNo one_third) H13R Hr2sRx Hm13R HloI3).
L174968
We prove the intermediate claim H13S: SNo one_third.
L174970
An exact proof term for the current goal is (real_SNo one_third H13R).
L174970
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L174972
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L174972
An exact proof term for the current goal is H0le_tmp.
L174973
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L174975
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L174975
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) (minus_SNo one_third)).
L174977
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r2s x) (minus_SNo one_third)) Hm23le0 H0le).
L174979
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r2s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L174982
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) 1 (minus_SNo one_third) Hr2sRx real_1 Hm13R HhiI3).
L174983
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L174985
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L174985
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L174986
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) (minus_SNo one_third)) two_thirds.
L174988
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L174991
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) (minus_SNo one_third)) Hm23R H23R Hr3Rx Hlow Hup).
L174995
Assume HxnotC3: ¬ (x ∈ C3).
L174995
We prove the intermediate claim HxX: x ∈ X.
L174997
An exact proof term for the current goal is (HAsubX x HxA).
L174997
We prove the intermediate claim HnotI1: ¬ (apply_fun r2s x ∈ I1).
L174999
Assume Hr2sI1': apply_fun r2s x ∈ I1.
L174999
We prove the intermediate claim Hr2sI1I: apply_fun r2s x ∈ I1 ∩ I.
L175001
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r2s x) Hr2sI1' Hr2sIx).
L175001
We prove the intermediate claim HxB3': x ∈ B3.
L175003
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r2s x0 ∈ I1 ∩ I) x HxA Hr2sI1I).
L175003
Apply FalseE to the current goal.
L175004
An exact proof term for the current goal is (HxnotB3 HxB3').
L175005
We prove the intermediate claim HnotI3: ¬ (apply_fun r2s x ∈ I3).
L175007
Assume Hr2sI3': apply_fun r2s x ∈ I3.
L175007
We prove the intermediate claim Hr2sI3I: apply_fun r2s x ∈ I3 ∩ I.
L175009
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r2s x) Hr2sI3' Hr2sIx).
L175009
We prove the intermediate claim HxC3': x ∈ C3.
L175011
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r2s x0 ∈ I3 ∩ I) x HxA Hr2sI3I).
L175011
Apply FalseE to the current goal.
L175012
An exact proof term for the current goal is (HxnotC3 HxC3').
L175013
We prove the intermediate claim Hr2sRx: apply_fun r2s x ∈ R.
L175015
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r2s x) Hr2sIx).
L175015
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175017
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175017
We prove the intermediate claim H13R: one_third ∈ R.
L175019
An exact proof term for the current goal is one_third_in_R.
L175019
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175021
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175021
We prove the intermediate claim Hr2s_boundsI: Rle (minus_SNo 1) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) 1.
L175023
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r2s x) Hm1R real_1 Hr2sIx).
L175023
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r2s x) (minus_SNo 1)).
L175025
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r2s x) (andEL (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_boundsI)).
L175026
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r2s x)).
L175028
An exact proof term for the current goal is (RleE_nlt (apply_fun r2s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r2s x)) (Rle (apply_fun r2s x) 1) Hr2s_boundsI)).
L175029
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r2s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r2s x).
L175031
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r2s x) Hm1R Hm13R Hr2sRx HnotI1).
L175032
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r2s x).
L175034
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r2s x))) to the current goal.
L175035
Assume Hbad: Rlt (apply_fun r2s x) (minus_SNo 1).
L175035
Apply FalseE to the current goal.
L175036
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L175038
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r2s x).
L175038
An exact proof term for the current goal is Hok.
L175039
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r2s x) (minus_SNo one_third)).
L175041
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r2s x) Hm13lt_fx).
L175041
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r2s x) one_third ∨ Rlt 1 (apply_fun r2s x).
L175043
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r2s x) H13R real_1 Hr2sRx HnotI3).
L175044
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r2s x) one_third.
L175046
Apply (HnotI3_cases (Rlt (apply_fun r2s x) one_third)) to the current goal.
L175047
Assume Hok: Rlt (apply_fun r2s x) one_third.
L175047
An exact proof term for the current goal is Hok.
L175049
Assume Hbad: Rlt 1 (apply_fun r2s x).
L175049
Apply FalseE to the current goal.
L175050
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L175051
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r2s x)).
L175053
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r2s x) one_third Hfx_lt_13).
L175053
We prove the intermediate claim Hr2sI0: apply_fun r2s x ∈ I0.
L175055
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L175057
We prove the intermediate claim HxSep: apply_fun r2s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L175059
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r2s x) Hr2sRx (andI (¬ (Rlt (apply_fun r2s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r2s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L175063
rewrite the current goal using HI0_def (from left to right).
L175064
An exact proof term for the current goal is HxSep.
L175065
We prove the intermediate claim Hu3funI0: function_on u3 X I0.
L175067
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u3 Hu3contI0).
L175067
We prove the intermediate claim Hu3xI0: apply_fun u3 x ∈ I0.
L175069
An exact proof term for the current goal is (Hu3funI0 x HxX).
L175069
We prove the intermediate claim Hu3xR: apply_fun u3 x ∈ R.
L175071
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u3 x) Hu3xI0).
L175071
We prove the intermediate claim Hm_u3x_R: minus_SNo (apply_fun u3 x) ∈ R.
L175073
An exact proof term for the current goal is (real_minus_SNo (apply_fun u3 x) Hu3xR).
L175073
rewrite the current goal using (Hr3_apply x HxA) (from left to right).
L175074
We prove the intermediate claim Hr3xR: add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) ∈ R.
L175076
An exact proof term for the current goal is (real_add_SNo (apply_fun r2s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) (minus_SNo (apply_fun u3 x)) Hm_u3x_R).
L175078
We prove the intermediate claim H23R: two_thirds ∈ R.
L175080
An exact proof term for the current goal is two_thirds_in_R.
L175080
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175082
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175082
We prove the intermediate claim Hr2s_bounds0: Rle (minus_SNo one_third) (apply_fun r2s x) ∧ Rle (apply_fun r2s x) one_third.
L175084
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r2s x) Hm13R H13R Hr2sI0).
L175084
We prove the intermediate claim Hu3_bounds0: Rle (minus_SNo one_third) (apply_fun u3 x) ∧ Rle (apply_fun u3 x) one_third.
L175086
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u3 x) Hm13R H13R Hu3xI0).
L175086
We prove the intermediate claim Hm13_le_r2s: Rle (minus_SNo one_third) (apply_fun r2s x).
L175088
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r2s x)) (Rle (apply_fun r2s x) one_third) Hr2s_bounds0).
L175088
We prove the intermediate claim Hr2s_le_13: Rle (apply_fun r2s x) one_third.
L175090
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r2s x)) (Rle (apply_fun r2s x) one_third) Hr2s_bounds0).
L175090
We prove the intermediate claim Hm13_le_u3x: Rle (minus_SNo one_third) (apply_fun u3 x).
L175092
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u3 x)) (Rle (apply_fun u3 x) one_third) Hu3_bounds0).
L175092
We prove the intermediate claim Hu3x_le_13: Rle (apply_fun u3 x) one_third.
L175094
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u3 x)) (Rle (apply_fun u3 x) one_third) Hu3_bounds0).
L175094
We prove the intermediate claim Hm13_le_mu3: Rle (minus_SNo one_third) (minus_SNo (apply_fun u3 x)).
L175096
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u3 x) one_third Hu3x_le_13).
L175096
We prove the intermediate claim Hmu3_le_13: Rle (minus_SNo (apply_fun u3 x)) one_third.
L175098
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u3 x)) (minus_SNo (minus_SNo one_third)).
L175099
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u3 x) Hm13_le_u3x).
L175099
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L175100
An exact proof term for the current goal is Htmp.
L175101
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))).
L175104
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u3 x)) Hm13R Hm13R Hm_u3x_R Hm13_le_mu3).
L175105
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L175108
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) Hm_u3x_R Hm13_le_r2s).
L175112
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L175115
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) Hlow1 Hlow2).
L175118
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))).
L175120
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L175120
An exact proof term for the current goal is Hlow_tmp.
L175121
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) one_third).
L175124
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r2s x) (minus_SNo (apply_fun u3 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) Hm_u3x_R H13R Hmu3_le_13).
L175126
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r2s x) one_third) (add_SNo one_third one_third).
L175129
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r2s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r2s x) Hr2sI0) H13R H13R Hr2s_le_13).
L175131
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo one_third one_third).
L175134
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) (add_SNo (apply_fun r2s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L175137
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L175139
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) two_thirds.
L175141
rewrite the current goal using Hdef23 (from left to right) at position 1.
L175141
An exact proof term for the current goal is Hup_tmp.
L175142
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r2s x) (minus_SNo (apply_fun u3 x))) Hm23R H23R Hr3xR Hlow Hup).
L175145
Set r3s to be the term compose_fun A r3 (div_const_fun den).
L175146
We prove the intermediate claim Hr3s_cont: continuous_map A Ta I Ti r3s.
L175148
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L175149
An exact proof term for the current goal is R_standard_topology_is_topology.
L175149
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L175151
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L175151
We prove the intermediate claim Hr3s_contR: continuous_map A Ta R R_standard_topology r3s.
L175153
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r3 (div_const_fun den) Hr3_cont Hdivcont).
L175154
We prove the intermediate claim Hr3s_I: ∀x : set, x ∈ A → apply_fun r3s x ∈ I.
L175156
Let x be given.
L175156
Assume HxA: x ∈ A.
L175156
We prove the intermediate claim Hr3xI2: apply_fun r3 x ∈ I2.
L175158
An exact proof term for the current goal is (Hr3_range x HxA).
L175158
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L175160
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175160
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L175162
An exact proof term for the current goal is (real_minus_SNo den H23R).
L175162
We prove the intermediate claim Hr3xR: apply_fun r3 x ∈ R.
L175164
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r3 x) Hr3xI2).
L175164
We prove the intermediate claim Hr3xS: SNo (apply_fun r3 x).
L175166
An exact proof term for the current goal is (real_SNo (apply_fun r3 x) Hr3xR).
L175166
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r3 x) ∧ Rle (apply_fun r3 x) den.
L175168
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r3 x) HmdenR H23R Hr3xI2).
L175168
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r3 x).
L175170
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r3 x)) (Rle (apply_fun r3 x) den) Hbounds).
L175172
We prove the intermediate claim Hhi: Rle (apply_fun r3 x) den.
L175174
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r3 x)) (Rle (apply_fun r3 x) den) Hbounds).
L175176
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r3 x)).
L175178
An exact proof term for the current goal is (RleE_nlt (apply_fun r3 x) den Hhi).
L175178
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r3 x) (minus_SNo den)).
L175180
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r3 x) Hlo).
L175180
We prove the intermediate claim HyEq: apply_fun r3s x = div_SNo (apply_fun r3 x) den.
L175182
rewrite the current goal using (compose_fun_apply A r3 (div_const_fun den) x HxA) (from left to right).
L175182
rewrite the current goal using (div_const_fun_apply den (apply_fun r3 x) H23R Hr3xR) (from left to right).
Use reflexivity.
L175184
We prove the intermediate claim HyR: apply_fun r3s x ∈ R.
L175186
rewrite the current goal using HyEq (from left to right).
L175186
An exact proof term for the current goal is (real_div_SNo (apply_fun r3 x) Hr3xR den H23R).
L175187
We prove the intermediate claim HyS: SNo (apply_fun r3s x).
L175189
An exact proof term for the current goal is (real_SNo (apply_fun r3s x) HyR).
L175189
We prove the intermediate claim Hy_le_1: Rle (apply_fun r3s x) 1.
L175191
Apply (RleI (apply_fun r3s x) 1 HyR real_1) to the current goal.
L175191
We will prove ¬ (Rlt 1 (apply_fun r3s x)).
L175192
Assume H1lt: Rlt 1 (apply_fun r3s x).
L175193
We prove the intermediate claim H1lty: 1 < apply_fun r3s x.
L175195
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r3s x) H1lt).
L175195
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r3s x) den.
L175197
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r3s x) den SNo_1 HyS H23S H23pos H1lty).
L175197
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r3s x) den.
L175199
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L175199
An exact proof term for the current goal is HmulLt.
L175200
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r3s x) den = apply_fun r3 x.
L175202
rewrite the current goal using HyEq (from left to right) at position 1.
L175202
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r3 x) den Hr3xS H23S H23ne0).
L175203
We prove the intermediate claim Hden_lt_r3x: den < apply_fun r3 x.
L175205
rewrite the current goal using HmulEq (from right to left).
L175205
An exact proof term for the current goal is HmulLt'.
L175206
We prove the intermediate claim Hbad: Rlt den (apply_fun r3 x).
L175208
An exact proof term for the current goal is (RltI den (apply_fun r3 x) H23R Hr3xR Hden_lt_r3x).
L175208
An exact proof term for the current goal is (Hnlt_hi Hbad).
L175209
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r3s x).
L175211
Apply (RleI (minus_SNo 1) (apply_fun r3s x) Hm1R HyR) to the current goal.
L175211
We will prove ¬ (Rlt (apply_fun r3s x) (minus_SNo 1)).
L175212
Assume Hylt: Rlt (apply_fun r3s x) (minus_SNo 1).
L175213
We prove the intermediate claim Hylts: apply_fun r3s x < minus_SNo 1.
L175215
An exact proof term for the current goal is (RltE_lt (apply_fun r3s x) (minus_SNo 1) Hylt).
L175215
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r3s x) den < mul_SNo (minus_SNo 1) den.
L175217
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r3s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L175218
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r3s x) den = apply_fun r3 x.
L175220
rewrite the current goal using HyEq (from left to right) at position 1.
L175220
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r3 x) den Hr3xS H23S H23ne0).
L175221
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L175223
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L175223
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L175225
We prove the intermediate claim Hr3x_lt_mden: apply_fun r3 x < minus_SNo den.
L175227
rewrite the current goal using HmulEq (from right to left).
L175227
rewrite the current goal using HrhsEq (from right to left).
L175228
An exact proof term for the current goal is HmulLt.
L175229
We prove the intermediate claim Hbad: Rlt (apply_fun r3 x) (minus_SNo den).
L175231
An exact proof term for the current goal is (RltI (apply_fun r3 x) (minus_SNo den) Hr3xR HmdenR Hr3x_lt_mden).
L175231
An exact proof term for the current goal is (Hnlt_lo Hbad).
L175232
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r3s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L175234
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r3s I Hr3s_contR HIcR Hr3s_I).
L175235
We prove the intermediate claim Hex_u4: ∃u4 : set, continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third).
(*** fifth correction step scaffold: build u4 and g4 from residual r3s ***)
L175245
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r3s Hnorm HA Hr3s_cont).
L175245
Apply Hex_u4 to the current goal.
L175246
Let u4 be given.
L175247
Assume Hu4.
L175247
We prove the intermediate claim Hu4contI0: continuous_map X Tx I0 T0 u4.
L175249
We prove the intermediate claim Hu4AB: continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third).
L175253
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L175259
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4) (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) Hu4AB).
L175264
We prove the intermediate claim Hu4contR: continuous_map X Tx R R_standard_topology u4.
L175266
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L175267
We prove the intermediate claim HI0subR: I0 ⊆ R.
L175269
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L175269
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u4 Hu4contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L175277
Set den4 to be the term mul_SNo den3 den.
L175278
We prove the intermediate claim Hden4R: den4 ∈ R.
L175280
An exact proof term for the current goal is (real_mul_SNo den3 Hden3R den H23R).
L175280
We prove the intermediate claim Hden4pos: 0 < den4.
L175282
We prove the intermediate claim Hden3S: SNo den3.
L175283
An exact proof term for the current goal is (real_SNo den3 Hden3R).
L175283
An exact proof term for the current goal is (mul_SNo_pos_pos den3 den Hden3S H23S Hden3pos HdenPos).
L175284
Set u4s to be the term compose_fun X u4 (mul_const_fun den4).
L175285
We prove the intermediate claim Hu4s_cont: continuous_map X Tx R R_standard_topology u4s.
L175287
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u4 den4 HTx Hu4contR Hden4R Hden4pos).
L175287
Set g4 to be the term compose_fun X (pair_map X g3 u4s) add_fun_R.
L175288
We prove the intermediate claim Hg4cont: continuous_map X Tx R R_standard_topology g4.
L175290
An exact proof term for the current goal is (add_two_continuous_R X Tx g3 u4s HTx Hg3cont Hu4s_cont).
L175290
We prove the intermediate claim Hu4contA: continuous_map A Ta R R_standard_topology u4.
(*** sixth correction step scaffold: residual r4 on A and scaling r4s ***)
L175294
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u4 A HTx HAsubX Hu4contR).
L175294
Set u4neg to be the term compose_fun A u4 neg_fun.
L175295
We prove the intermediate claim Hu4neg_cont: continuous_map A Ta R R_standard_topology u4neg.
L175297
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u4 neg_fun Hu4contA Hnegcont).
L175298
We prove the intermediate claim Hr3s_contR: continuous_map A Ta R R_standard_topology r3s.
L175300
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r3s Hr3s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L175301
Set r4 to be the term compose_fun A (pair_map A r3s u4neg) add_fun_R.
L175302
We prove the intermediate claim Hr4_cont: continuous_map A Ta R R_standard_topology r4.
L175304
An exact proof term for the current goal is (add_two_continuous_R A Ta r3s u4neg HTa Hr3s_contR Hu4neg_cont).
L175304
We prove the intermediate claim Hr4_apply: ∀x : set, x ∈ A → apply_fun r4 x = add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x)).
L175307
Let x be given.
L175307
Assume HxA: x ∈ A.
L175307
We prove the intermediate claim Hpimg: apply_fun (pair_map A r3s u4neg) x ∈ setprod R R.
L175309
rewrite the current goal using (pair_map_apply A R R r3s u4neg x HxA) (from left to right).
L175309
We prove the intermediate claim Hr3sxI: apply_fun r3s x ∈ I.
L175311
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r3s Hr3s_cont x HxA).
L175311
We prove the intermediate claim Hr3sxR: apply_fun r3s x ∈ R.
L175313
An exact proof term for the current goal is (HIcR (apply_fun r3s x) Hr3sxI).
L175313
We prove the intermediate claim Hu4negRx: apply_fun u4neg x ∈ R.
L175315
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u4neg Hu4neg_cont x HxA).
L175315
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r3s x) (apply_fun u4neg x) Hr3sxR Hu4negRx).
L175316
rewrite the current goal using (compose_fun_apply A (pair_map A r3s u4neg) add_fun_R x HxA) (from left to right).
L175317
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r3s u4neg) x) Hpimg) (from left to right) at position 1.
L175318
rewrite the current goal using (pair_map_apply A R R r3s u4neg x HxA) (from left to right).
L175319
rewrite the current goal using (tuple_2_0_eq (apply_fun r3s x) (apply_fun u4neg x)) (from left to right).
L175320
rewrite the current goal using (tuple_2_1_eq (apply_fun r3s x) (apply_fun u4neg x)) (from left to right).
L175321
rewrite the current goal using (compose_fun_apply A u4 neg_fun x HxA) (from left to right) at position 1.
L175322
We prove the intermediate claim Hu4Rx: apply_fun u4 x ∈ R.
L175324
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u4 Hu4contA x HxA).
L175324
rewrite the current goal using (neg_fun_apply (apply_fun u4 x) Hu4Rx) (from left to right) at position 1.
Use reflexivity.
L175326
We prove the intermediate claim Hr4_range: ∀x : set, x ∈ A → apply_fun r4 x ∈ I2.
L175328
We prove the intermediate claim Hu4AB: continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L175333
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L175339
We prove the intermediate claim Hu4_on_B4: ∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third.
L175343
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u4) (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third) Hu4AB).
L175347
We prove the intermediate claim Hu4_on_C4: ∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third.
L175351
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u4 ∧ (∀x : set, x ∈ preimage_of A r3s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u4 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r3s ((closed_interval one_third 1) ∩ I) → apply_fun u4 x = one_third) Hu4).
L175357
Let x be given.
L175358
Assume HxA: x ∈ A.
L175358
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L175359
Set I3 to be the term closed_interval one_third 1.
L175360
Set B4 to be the term preimage_of A r3s (I1 ∩ I).
L175361
Set C4 to be the term preimage_of A r3s (I3 ∩ I).
L175362
We prove the intermediate claim Hr3sIx: apply_fun r3s x ∈ I.
L175364
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r3s Hr3s_cont x HxA).
L175364
We prove the intermediate claim HB4_cases: x ∈ B4 ∨ ¬ (x ∈ B4).
L175366
An exact proof term for the current goal is (xm (x ∈ B4)).
L175366
Apply (HB4_cases (apply_fun r4 x ∈ I2)) to the current goal.
L175368
Assume HxB4: x ∈ B4.
L175368
We prove the intermediate claim Hu4eq: apply_fun u4 x = minus_SNo one_third.
L175370
An exact proof term for the current goal is (Hu4_on_B4 x HxB4).
L175370
We prove the intermediate claim Hr4eq: apply_fun r4 x = add_SNo (apply_fun r3s x) one_third.
L175372
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L175372
rewrite the current goal using Hu4eq (from left to right) at position 1.
L175373
We prove the intermediate claim H13R: one_third ∈ R.
L175375
An exact proof term for the current goal is one_third_in_R.
L175375
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L175377
rewrite the current goal using Hr4eq (from left to right).
L175378
We prove the intermediate claim Hr3sI1I: apply_fun r3s x ∈ I1 ∩ I.
L175380
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r3s x0 ∈ I1 ∩ I) x HxB4).
L175380
We prove the intermediate claim Hr3sI1: apply_fun r3s x ∈ I1.
L175382
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r3s x) Hr3sI1I).
L175382
We prove the intermediate claim H13R: one_third ∈ R.
L175384
An exact proof term for the current goal is one_third_in_R.
L175384
We prove the intermediate claim H23R: two_thirds ∈ R.
L175386
An exact proof term for the current goal is two_thirds_in_R.
L175386
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175388
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175388
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175390
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175390
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175392
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175392
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L175394
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hr3sI1).
L175394
We prove the intermediate claim Hr3s_bounds: Rle (minus_SNo 1) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) (minus_SNo one_third).
L175396
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hm1R Hm13R Hr3sI1).
L175397
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r3s x).
L175399
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) (minus_SNo one_third)) Hr3s_bounds).
L175400
We prove the intermediate claim HhiI1: Rle (apply_fun r3s x) (minus_SNo one_third).
L175402
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) (minus_SNo one_third)) Hr3s_bounds).
L175403
We prove the intermediate claim Hr4Rx: add_SNo (apply_fun r3s x) one_third ∈ R.
L175405
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) Hr3sRx one_third H13R).
L175405
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r3s x) one_third).
L175407
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r3s x) one_third Hm1R Hr3sRx H13R Hm1le).
L175407
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) one_third).
L175409
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L175409
An exact proof term for the current goal is Hlow_tmp.
L175410
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r3s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L175412
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) (minus_SNo one_third) one_third Hr3sRx Hm13R H13R HhiI1).
L175412
We prove the intermediate claim H13S: SNo one_third.
L175414
An exact proof term for the current goal is (real_SNo one_third H13R).
L175414
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r3s x) one_third) 0.
L175416
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L175416
An exact proof term for the current goal is Hup0_tmp.
L175417
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L175419
An exact proof term for the current goal is Rle_0_two_thirds.
L175419
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) one_third) two_thirds.
L175421
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) one_third) 0 two_thirds Hup0 H0le23).
L175421
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) one_third) Hm23R H23R Hr4Rx Hlow Hup).
L175425
Assume HxnotB4: ¬ (x ∈ B4).
L175425
We prove the intermediate claim HC4_cases: x ∈ C4 ∨ ¬ (x ∈ C4).
L175427
An exact proof term for the current goal is (xm (x ∈ C4)).
L175427
Apply (HC4_cases (apply_fun r4 x ∈ I2)) to the current goal.
L175429
Assume HxC4: x ∈ C4.
L175429
We prove the intermediate claim Hu4eq: apply_fun u4 x = one_third.
L175431
An exact proof term for the current goal is (Hu4_on_C4 x HxC4).
L175431
We prove the intermediate claim Hr4eq: apply_fun r4 x = add_SNo (apply_fun r3s x) (minus_SNo one_third).
L175433
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L175433
rewrite the current goal using Hu4eq (from left to right) at position 1.
Use reflexivity.
L175435
rewrite the current goal using Hr4eq (from left to right).
L175436
We prove the intermediate claim Hr3sI3I: apply_fun r3s x ∈ I3 ∩ I.
L175438
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r3s x0 ∈ I3 ∩ I) x HxC4).
L175438
We prove the intermediate claim Hr3sI3: apply_fun r3s x ∈ I3.
L175440
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r3s x) Hr3sI3I).
L175440
We prove the intermediate claim H13R: one_third ∈ R.
L175442
An exact proof term for the current goal is one_third_in_R.
L175442
We prove the intermediate claim H23R: two_thirds ∈ R.
L175444
An exact proof term for the current goal is two_thirds_in_R.
L175444
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175446
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175446
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175448
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175448
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L175450
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r3s x) Hr3sI3).
L175450
We prove the intermediate claim Hr3s_bounds: Rle one_third (apply_fun r3s x) ∧ Rle (apply_fun r3s x) 1.
L175452
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r3s x) H13R real_1 Hr3sI3).
L175452
We prove the intermediate claim HloI3: Rle one_third (apply_fun r3s x).
L175454
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_bounds).
L175454
We prove the intermediate claim HhiI3: Rle (apply_fun r3s x) 1.
L175456
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_bounds).
L175456
We prove the intermediate claim Hr4Rx: add_SNo (apply_fun r3s x) (minus_SNo one_third) ∈ R.
L175458
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) Hr3sRx (minus_SNo one_third) Hm13R).
L175458
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L175461
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r3s x) (minus_SNo one_third) H13R Hr3sRx Hm13R HloI3).
L175462
We prove the intermediate claim H13S: SNo one_third.
L175464
An exact proof term for the current goal is (real_SNo one_third H13R).
L175464
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L175466
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L175466
An exact proof term for the current goal is H0le_tmp.
L175467
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L175469
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L175469
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) (minus_SNo one_third)).
L175471
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r3s x) (minus_SNo one_third)) Hm23le0 H0le).
L175473
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r3s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L175476
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) 1 (minus_SNo one_third) Hr3sRx real_1 Hm13R HhiI3).
L175477
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L175479
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L175479
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L175480
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) (minus_SNo one_third)) two_thirds.
L175482
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L175485
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) (minus_SNo one_third)) Hm23R H23R Hr4Rx Hlow Hup).
L175489
Assume HxnotC4: ¬ (x ∈ C4).
L175489
We prove the intermediate claim HxX: x ∈ X.
L175491
An exact proof term for the current goal is (HAsubX x HxA).
L175491
We prove the intermediate claim HnotI1: ¬ (apply_fun r3s x ∈ I1).
L175493
Assume Hr3sI1': apply_fun r3s x ∈ I1.
L175493
We prove the intermediate claim Hr3sI1I: apply_fun r3s x ∈ I1 ∩ I.
L175495
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r3s x) Hr3sI1' Hr3sIx).
L175495
We prove the intermediate claim HxB4': x ∈ B4.
L175497
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r3s x0 ∈ I1 ∩ I) x HxA Hr3sI1I).
L175497
Apply FalseE to the current goal.
L175498
An exact proof term for the current goal is (HxnotB4 HxB4').
L175499
We prove the intermediate claim HnotI3: ¬ (apply_fun r3s x ∈ I3).
L175501
Assume Hr3sI3': apply_fun r3s x ∈ I3.
L175501
We prove the intermediate claim Hr3sI3I: apply_fun r3s x ∈ I3 ∩ I.
L175503
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r3s x) Hr3sI3' Hr3sIx).
L175503
We prove the intermediate claim HxC4': x ∈ C4.
L175505
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r3s x0 ∈ I3 ∩ I) x HxA Hr3sI3I).
L175505
Apply FalseE to the current goal.
L175506
An exact proof term for the current goal is (HxnotC4 HxC4').
L175507
We prove the intermediate claim Hr3sRx: apply_fun r3s x ∈ R.
L175509
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r3s x) Hr3sIx).
L175509
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175511
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175511
We prove the intermediate claim H13R: one_third ∈ R.
L175513
An exact proof term for the current goal is one_third_in_R.
L175513
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175515
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175515
We prove the intermediate claim Hr3s_boundsI: Rle (minus_SNo 1) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) 1.
L175517
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r3s x) Hm1R real_1 Hr3sIx).
L175517
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r3s x) (minus_SNo 1)).
L175519
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r3s x) (andEL (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_boundsI)).
L175520
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r3s x)).
L175522
An exact proof term for the current goal is (RleE_nlt (apply_fun r3s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r3s x)) (Rle (apply_fun r3s x) 1) Hr3s_boundsI)).
L175523
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r3s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r3s x).
L175525
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r3s x) Hm1R Hm13R Hr3sRx HnotI1).
L175526
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r3s x).
L175528
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r3s x))) to the current goal.
L175529
Assume Hbad: Rlt (apply_fun r3s x) (minus_SNo 1).
L175529
Apply FalseE to the current goal.
L175530
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L175532
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r3s x).
L175532
An exact proof term for the current goal is Hok.
L175533
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r3s x) (minus_SNo one_third)).
L175535
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r3s x) Hm13lt_fx).
L175535
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r3s x) one_third ∨ Rlt 1 (apply_fun r3s x).
L175537
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r3s x) H13R real_1 Hr3sRx HnotI3).
L175538
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r3s x) one_third.
L175540
Apply (HnotI3_cases (Rlt (apply_fun r3s x) one_third)) to the current goal.
L175541
Assume Hok: Rlt (apply_fun r3s x) one_third.
L175541
An exact proof term for the current goal is Hok.
L175543
Assume Hbad: Rlt 1 (apply_fun r3s x).
L175543
Apply FalseE to the current goal.
L175544
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L175545
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r3s x)).
L175547
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r3s x) one_third Hfx_lt_13).
L175547
We prove the intermediate claim Hr3sI0: apply_fun r3s x ∈ I0.
L175549
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L175551
We prove the intermediate claim HxSep: apply_fun r3s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L175553
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r3s x) Hr3sRx (andI (¬ (Rlt (apply_fun r3s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r3s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L175557
rewrite the current goal using HI0_def (from left to right).
L175558
An exact proof term for the current goal is HxSep.
L175559
We prove the intermediate claim Hu4funI0: function_on u4 X I0.
L175561
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u4 Hu4contI0).
L175561
We prove the intermediate claim Hu4xI0: apply_fun u4 x ∈ I0.
L175563
An exact proof term for the current goal is (Hu4funI0 x HxX).
L175563
We prove the intermediate claim Hu4xR: apply_fun u4 x ∈ R.
L175565
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u4 x) Hu4xI0).
L175565
We prove the intermediate claim Hm_u4x_R: minus_SNo (apply_fun u4 x) ∈ R.
L175567
An exact proof term for the current goal is (real_minus_SNo (apply_fun u4 x) Hu4xR).
L175567
rewrite the current goal using (Hr4_apply x HxA) (from left to right).
L175568
We prove the intermediate claim Hr4xR: add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) ∈ R.
L175570
An exact proof term for the current goal is (real_add_SNo (apply_fun r3s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) (minus_SNo (apply_fun u4 x)) Hm_u4x_R).
L175572
We prove the intermediate claim H23R: two_thirds ∈ R.
L175574
An exact proof term for the current goal is two_thirds_in_R.
L175574
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175576
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175576
We prove the intermediate claim Hr3s_bounds0: Rle (minus_SNo one_third) (apply_fun r3s x) ∧ Rle (apply_fun r3s x) one_third.
L175578
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r3s x) Hm13R H13R Hr3sI0).
L175578
We prove the intermediate claim Hu4_bounds0: Rle (minus_SNo one_third) (apply_fun u4 x) ∧ Rle (apply_fun u4 x) one_third.
L175580
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u4 x) Hm13R H13R Hu4xI0).
L175580
We prove the intermediate claim Hm13_le_r3s: Rle (minus_SNo one_third) (apply_fun r3s x).
L175582
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r3s x)) (Rle (apply_fun r3s x) one_third) Hr3s_bounds0).
L175582
We prove the intermediate claim Hr3s_le_13: Rle (apply_fun r3s x) one_third.
L175584
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r3s x)) (Rle (apply_fun r3s x) one_third) Hr3s_bounds0).
L175584
We prove the intermediate claim Hm13_le_u4x: Rle (minus_SNo one_third) (apply_fun u4 x).
L175586
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u4 x)) (Rle (apply_fun u4 x) one_third) Hu4_bounds0).
L175586
We prove the intermediate claim Hu4x_le_13: Rle (apply_fun u4 x) one_third.
L175588
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u4 x)) (Rle (apply_fun u4 x) one_third) Hu4_bounds0).
L175588
We prove the intermediate claim Hm13_le_mu4: Rle (minus_SNo one_third) (minus_SNo (apply_fun u4 x)).
L175590
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u4 x) one_third Hu4x_le_13).
L175590
We prove the intermediate claim Hmu4_le_13: Rle (minus_SNo (apply_fun u4 x)) one_third.
L175592
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u4 x)) (minus_SNo (minus_SNo one_third)).
L175593
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u4 x) Hm13_le_u4x).
L175593
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L175594
An exact proof term for the current goal is Htmp.
L175595
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))).
L175598
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u4 x)) Hm13R Hm13R Hm_u4x_R Hm13_le_mu4).
L175599
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L175602
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) Hm_u4x_R Hm13_le_r3s).
L175606
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L175609
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) Hlow1 Hlow2).
L175612
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))).
L175614
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L175614
An exact proof term for the current goal is Hlow_tmp.
L175615
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) one_third).
L175618
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r3s x) (minus_SNo (apply_fun u4 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) Hm_u4x_R H13R Hmu4_le_13).
L175620
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r3s x) one_third) (add_SNo one_third one_third).
L175623
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r3s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r3s x) Hr3sI0) H13R H13R Hr3s_le_13).
L175625
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo one_third one_third).
L175628
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) (add_SNo (apply_fun r3s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L175631
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L175633
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) two_thirds.
L175635
rewrite the current goal using Hdef23 (from left to right) at position 1.
L175635
An exact proof term for the current goal is Hup_tmp.
L175636
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r3s x) (minus_SNo (apply_fun u4 x))) Hm23R H23R Hr4xR Hlow Hup).
L175639
Set r4s to be the term compose_fun A r4 (div_const_fun den).
L175640
We prove the intermediate claim Hr4s_cont: continuous_map A Ta I Ti r4s.
L175642
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L175643
An exact proof term for the current goal is R_standard_topology_is_topology.
L175643
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L175645
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L175645
We prove the intermediate claim Hr4s_contR: continuous_map A Ta R R_standard_topology r4s.
L175647
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r4 (div_const_fun den) Hr4_cont Hdivcont).
L175648
We prove the intermediate claim Hr4s_I: ∀x : set, x ∈ A → apply_fun r4s x ∈ I.
L175650
Let x be given.
L175650
Assume HxA: x ∈ A.
L175650
We prove the intermediate claim Hr4xI2: apply_fun r4 x ∈ I2.
L175652
An exact proof term for the current goal is (Hr4_range x HxA).
L175652
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L175654
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175654
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L175656
An exact proof term for the current goal is (real_minus_SNo den H23R).
L175656
We prove the intermediate claim Hr4xR: apply_fun r4 x ∈ R.
L175658
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r4 x) Hr4xI2).
L175658
We prove the intermediate claim Hr4xS: SNo (apply_fun r4 x).
L175660
An exact proof term for the current goal is (real_SNo (apply_fun r4 x) Hr4xR).
L175660
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r4 x) ∧ Rle (apply_fun r4 x) den.
L175662
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r4 x) HmdenR H23R Hr4xI2).
L175662
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r4 x).
L175664
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r4 x)) (Rle (apply_fun r4 x) den) Hbounds).
L175666
We prove the intermediate claim Hhi: Rle (apply_fun r4 x) den.
L175668
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r4 x)) (Rle (apply_fun r4 x) den) Hbounds).
L175670
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r4 x)).
L175672
An exact proof term for the current goal is (RleE_nlt (apply_fun r4 x) den Hhi).
L175672
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r4 x) (minus_SNo den)).
L175674
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r4 x) Hlo).
L175674
We prove the intermediate claim HyEq: apply_fun r4s x = div_SNo (apply_fun r4 x) den.
L175676
rewrite the current goal using (compose_fun_apply A r4 (div_const_fun den) x HxA) (from left to right).
L175676
rewrite the current goal using (div_const_fun_apply den (apply_fun r4 x) H23R Hr4xR) (from left to right).
Use reflexivity.
L175678
We prove the intermediate claim HyR: apply_fun r4s x ∈ R.
L175680
rewrite the current goal using HyEq (from left to right).
L175680
An exact proof term for the current goal is (real_div_SNo (apply_fun r4 x) Hr4xR den H23R).
L175681
We prove the intermediate claim HyS: SNo (apply_fun r4s x).
L175683
An exact proof term for the current goal is (real_SNo (apply_fun r4s x) HyR).
L175683
We prove the intermediate claim Hy_le_1: Rle (apply_fun r4s x) 1.
L175685
Apply (RleI (apply_fun r4s x) 1 HyR real_1) to the current goal.
L175685
We will prove ¬ (Rlt 1 (apply_fun r4s x)).
L175686
Assume H1lt: Rlt 1 (apply_fun r4s x).
L175687
We prove the intermediate claim H1lty: 1 < apply_fun r4s x.
L175689
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r4s x) H1lt).
L175689
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r4s x) den.
L175691
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r4s x) den SNo_1 HyS H23S H23pos H1lty).
L175691
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r4s x) den.
L175693
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L175693
An exact proof term for the current goal is HmulLt.
L175694
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r4s x) den = apply_fun r4 x.
L175696
rewrite the current goal using HyEq (from left to right) at position 1.
L175696
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r4 x) den Hr4xS H23S H23ne0).
L175697
We prove the intermediate claim Hden_lt_r4x: den < apply_fun r4 x.
L175699
rewrite the current goal using HmulEq (from right to left).
L175699
An exact proof term for the current goal is HmulLt'.
L175700
We prove the intermediate claim Hbad: Rlt den (apply_fun r4 x).
L175702
An exact proof term for the current goal is (RltI den (apply_fun r4 x) H23R Hr4xR Hden_lt_r4x).
L175702
An exact proof term for the current goal is (Hnlt_hi Hbad).
L175703
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r4s x).
L175705
Apply (RleI (minus_SNo 1) (apply_fun r4s x) Hm1R HyR) to the current goal.
L175705
We will prove ¬ (Rlt (apply_fun r4s x) (minus_SNo 1)).
L175706
Assume Hylt: Rlt (apply_fun r4s x) (minus_SNo 1).
L175707
We prove the intermediate claim Hylts: apply_fun r4s x < minus_SNo 1.
L175709
An exact proof term for the current goal is (RltE_lt (apply_fun r4s x) (minus_SNo 1) Hylt).
L175709
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r4s x) den < mul_SNo (minus_SNo 1) den.
L175711
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r4s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L175712
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r4s x) den = apply_fun r4 x.
L175714
rewrite the current goal using HyEq (from left to right) at position 1.
L175714
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r4 x) den Hr4xS H23S H23ne0).
L175715
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L175717
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L175717
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L175719
We prove the intermediate claim Hr4x_lt_mden: apply_fun r4 x < minus_SNo den.
L175721
rewrite the current goal using HmulEq (from right to left).
L175721
rewrite the current goal using HrhsEq (from right to left).
L175722
An exact proof term for the current goal is HmulLt.
L175723
We prove the intermediate claim Hbad: Rlt (apply_fun r4 x) (minus_SNo den).
L175725
An exact proof term for the current goal is (RltI (apply_fun r4 x) (minus_SNo den) Hr4xR HmdenR Hr4x_lt_mden).
L175725
An exact proof term for the current goal is (Hnlt_lo Hbad).
L175726
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r4s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L175728
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r4s I Hr4s_contR HIcR Hr4s_I).
L175729
We prove the intermediate claim Hex_u5: ∃u5 : set, continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third).
(*** seventh correction step scaffold: build u5 and g5 from residual r4s ***)
L175739
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r4s Hnorm HA Hr4s_cont).
L175739
Apply Hex_u5 to the current goal.
L175740
Let u5 be given.
L175741
Assume Hu5.
L175741
We prove the intermediate claim Hu5contI0: continuous_map X Tx I0 T0 u5.
L175743
We prove the intermediate claim Hu5AB: continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third).
L175747
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L175753
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5) (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) Hu5AB).
L175758
We prove the intermediate claim Hu5contR: continuous_map X Tx R R_standard_topology u5.
L175760
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L175761
We prove the intermediate claim HI0subR: I0 ⊆ R.
L175763
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L175763
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u5 Hu5contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L175771
Set den5 to be the term mul_SNo den4 den.
L175772
We prove the intermediate claim Hden5R: den5 ∈ R.
L175774
An exact proof term for the current goal is (real_mul_SNo den4 Hden4R den H23R).
L175774
We prove the intermediate claim Hden5pos: 0 < den5.
L175776
We prove the intermediate claim Hden4S: SNo den4.
L175777
An exact proof term for the current goal is (real_SNo den4 Hden4R).
L175777
An exact proof term for the current goal is (mul_SNo_pos_pos den4 den Hden4S H23S Hden4pos HdenPos).
L175778
Set u5s to be the term compose_fun X u5 (mul_const_fun den5).
L175779
We prove the intermediate claim Hu5s_cont: continuous_map X Tx R R_standard_topology u5s.
L175781
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u5 den5 HTx Hu5contR Hden5R Hden5pos).
L175781
Set g5 to be the term compose_fun X (pair_map X g4 u5s) add_fun_R.
L175782
We prove the intermediate claim Hg5cont: continuous_map X Tx R R_standard_topology g5.
L175784
An exact proof term for the current goal is (add_two_continuous_R X Tx g4 u5s HTx Hg4cont Hu5s_cont).
L175784
We prove the intermediate claim Hu5contA: continuous_map A Ta R R_standard_topology u5.
(*** eighth correction step scaffold: residual r5 on A and scaling r5s ***)
L175788
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u5 A HTx HAsubX Hu5contR).
L175788
Set u5neg to be the term compose_fun A u5 neg_fun.
L175789
We prove the intermediate claim Hu5neg_cont: continuous_map A Ta R R_standard_topology u5neg.
L175791
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u5 neg_fun Hu5contA Hnegcont).
L175792
We prove the intermediate claim Hr4s_contR: continuous_map A Ta R R_standard_topology r4s.
L175794
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r4s Hr4s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L175795
Set r5 to be the term compose_fun A (pair_map A r4s u5neg) add_fun_R.
L175796
We prove the intermediate claim Hr5_cont: continuous_map A Ta R R_standard_topology r5.
L175798
An exact proof term for the current goal is (add_two_continuous_R A Ta r4s u5neg HTa Hr4s_contR Hu5neg_cont).
L175798
We prove the intermediate claim Hr5_apply: ∀x : set, x ∈ A → apply_fun r5 x = add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x)).
L175801
Let x be given.
L175801
Assume HxA: x ∈ A.
L175801
We prove the intermediate claim Hpimg: apply_fun (pair_map A r4s u5neg) x ∈ setprod R R.
L175803
rewrite the current goal using (pair_map_apply A R R r4s u5neg x HxA) (from left to right).
L175803
We prove the intermediate claim Hr4sxI: apply_fun r4s x ∈ I.
L175805
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r4s Hr4s_cont x HxA).
L175805
We prove the intermediate claim Hr4sxR: apply_fun r4s x ∈ R.
L175807
An exact proof term for the current goal is (HIcR (apply_fun r4s x) Hr4sxI).
L175807
We prove the intermediate claim Hu5negRx: apply_fun u5neg x ∈ R.
L175809
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u5neg Hu5neg_cont x HxA).
L175809
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r4s x) (apply_fun u5neg x) Hr4sxR Hu5negRx).
L175810
rewrite the current goal using (compose_fun_apply A (pair_map A r4s u5neg) add_fun_R x HxA) (from left to right).
L175811
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r4s u5neg) x) Hpimg) (from left to right) at position 1.
L175812
rewrite the current goal using (pair_map_apply A R R r4s u5neg x HxA) (from left to right).
L175813
rewrite the current goal using (tuple_2_0_eq (apply_fun r4s x) (apply_fun u5neg x)) (from left to right).
L175814
rewrite the current goal using (tuple_2_1_eq (apply_fun r4s x) (apply_fun u5neg x)) (from left to right).
L175815
rewrite the current goal using (compose_fun_apply A u5 neg_fun x HxA) (from left to right) at position 1.
L175816
We prove the intermediate claim Hu5Rx: apply_fun u5 x ∈ R.
L175818
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u5 Hu5contA x HxA).
L175818
rewrite the current goal using (neg_fun_apply (apply_fun u5 x) Hu5Rx) (from left to right) at position 1.
Use reflexivity.
L175820
We prove the intermediate claim Hr5_range: ∀x : set, x ∈ A → apply_fun r5 x ∈ I2.
L175822
We prove the intermediate claim Hu5AB: continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L175827
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L175833
We prove the intermediate claim Hu5_on_B5: ∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third.
L175837
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u5) (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third) Hu5AB).
L175841
We prove the intermediate claim Hu5_on_C5: ∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third.
L175845
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u5 ∧ (∀x : set, x ∈ preimage_of A r4s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u5 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r4s ((closed_interval one_third 1) ∩ I) → apply_fun u5 x = one_third) Hu5).
L175851
Let x be given.
L175852
Assume HxA: x ∈ A.
L175852
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L175853
Set I3 to be the term closed_interval one_third 1.
L175854
Set B5 to be the term preimage_of A r4s (I1 ∩ I).
L175855
Set C5 to be the term preimage_of A r4s (I3 ∩ I).
L175856
We prove the intermediate claim Hr4sIx: apply_fun r4s x ∈ I.
L175858
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r4s Hr4s_cont x HxA).
L175858
We prove the intermediate claim HB5_cases: x ∈ B5 ∨ ¬ (x ∈ B5).
L175860
An exact proof term for the current goal is (xm (x ∈ B5)).
L175860
Apply (HB5_cases (apply_fun r5 x ∈ I2)) to the current goal.
L175862
Assume HxB5: x ∈ B5.
L175862
We prove the intermediate claim Hu5eq: apply_fun u5 x = minus_SNo one_third.
L175864
An exact proof term for the current goal is (Hu5_on_B5 x HxB5).
L175864
We prove the intermediate claim Hr5eq: apply_fun r5 x = add_SNo (apply_fun r4s x) one_third.
L175866
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L175866
rewrite the current goal using Hu5eq (from left to right) at position 1.
L175867
We prove the intermediate claim H13R: one_third ∈ R.
L175869
An exact proof term for the current goal is one_third_in_R.
L175869
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L175871
rewrite the current goal using Hr5eq (from left to right).
L175872
We prove the intermediate claim Hr4sI1I: apply_fun r4s x ∈ I1 ∩ I.
L175874
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r4s x0 ∈ I1 ∩ I) x HxB5).
L175874
We prove the intermediate claim Hr4sI1: apply_fun r4s x ∈ I1.
L175876
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r4s x) Hr4sI1I).
L175876
We prove the intermediate claim H13R: one_third ∈ R.
L175878
An exact proof term for the current goal is one_third_in_R.
L175878
We prove the intermediate claim H23R: two_thirds ∈ R.
L175880
An exact proof term for the current goal is two_thirds_in_R.
L175880
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175882
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175882
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L175884
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L175884
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175886
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175886
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L175888
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hr4sI1).
L175888
We prove the intermediate claim Hr4s_bounds: Rle (minus_SNo 1) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) (minus_SNo one_third).
L175890
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hm1R Hm13R Hr4sI1).
L175891
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r4s x).
L175893
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) (minus_SNo one_third)) Hr4s_bounds).
L175894
We prove the intermediate claim HhiI1: Rle (apply_fun r4s x) (minus_SNo one_third).
L175896
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) (minus_SNo one_third)) Hr4s_bounds).
L175897
We prove the intermediate claim Hr5Rx: add_SNo (apply_fun r4s x) one_third ∈ R.
L175899
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) Hr4sRx one_third H13R).
L175899
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r4s x) one_third).
L175901
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r4s x) one_third Hm1R Hr4sRx H13R Hm1le).
L175901
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) one_third).
L175903
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L175903
An exact proof term for the current goal is Hlow_tmp.
L175904
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r4s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L175906
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) (minus_SNo one_third) one_third Hr4sRx Hm13R H13R HhiI1).
L175906
We prove the intermediate claim H13S: SNo one_third.
L175908
An exact proof term for the current goal is (real_SNo one_third H13R).
L175908
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r4s x) one_third) 0.
L175910
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L175910
An exact proof term for the current goal is Hup0_tmp.
L175911
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L175913
An exact proof term for the current goal is Rle_0_two_thirds.
L175913
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) one_third) two_thirds.
L175915
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) one_third) 0 two_thirds Hup0 H0le23).
L175915
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) one_third) Hm23R H23R Hr5Rx Hlow Hup).
L175919
Assume HxnotB5: ¬ (x ∈ B5).
L175919
We prove the intermediate claim HC5_cases: x ∈ C5 ∨ ¬ (x ∈ C5).
L175921
An exact proof term for the current goal is (xm (x ∈ C5)).
L175921
Apply (HC5_cases (apply_fun r5 x ∈ I2)) to the current goal.
L175923
Assume HxC5: x ∈ C5.
L175923
We prove the intermediate claim Hu5eq: apply_fun u5 x = one_third.
L175925
An exact proof term for the current goal is (Hu5_on_C5 x HxC5).
L175925
We prove the intermediate claim Hr5eq: apply_fun r5 x = add_SNo (apply_fun r4s x) (minus_SNo one_third).
L175927
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L175927
rewrite the current goal using Hu5eq (from left to right) at position 1.
Use reflexivity.
L175929
rewrite the current goal using Hr5eq (from left to right).
L175930
We prove the intermediate claim Hr4sI3I: apply_fun r4s x ∈ I3 ∩ I.
L175932
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r4s x0 ∈ I3 ∩ I) x HxC5).
L175932
We prove the intermediate claim Hr4sI3: apply_fun r4s x ∈ I3.
L175934
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r4s x) Hr4sI3I).
L175934
We prove the intermediate claim H13R: one_third ∈ R.
L175936
An exact proof term for the current goal is one_third_in_R.
L175936
We prove the intermediate claim H23R: two_thirds ∈ R.
L175938
An exact proof term for the current goal is two_thirds_in_R.
L175938
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L175940
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L175940
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L175942
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L175942
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L175944
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r4s x) Hr4sI3).
L175944
We prove the intermediate claim Hr4s_bounds: Rle one_third (apply_fun r4s x) ∧ Rle (apply_fun r4s x) 1.
L175946
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r4s x) H13R real_1 Hr4sI3).
L175946
We prove the intermediate claim HloI3: Rle one_third (apply_fun r4s x).
L175948
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_bounds).
L175948
We prove the intermediate claim HhiI3: Rle (apply_fun r4s x) 1.
L175950
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_bounds).
L175950
We prove the intermediate claim Hr5Rx: add_SNo (apply_fun r4s x) (minus_SNo one_third) ∈ R.
L175952
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) Hr4sRx (minus_SNo one_third) Hm13R).
L175952
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L175955
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r4s x) (minus_SNo one_third) H13R Hr4sRx Hm13R HloI3).
L175956
We prove the intermediate claim H13S: SNo one_third.
L175958
An exact proof term for the current goal is (real_SNo one_third H13R).
L175958
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L175960
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L175960
An exact proof term for the current goal is H0le_tmp.
L175961
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L175963
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L175963
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) (minus_SNo one_third)).
L175965
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r4s x) (minus_SNo one_third)) Hm23le0 H0le).
L175967
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r4s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L175970
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) 1 (minus_SNo one_third) Hr4sRx real_1 Hm13R HhiI3).
L175971
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L175973
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L175973
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L175974
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) (minus_SNo one_third)) two_thirds.
L175976
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L175979
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) (minus_SNo one_third)) Hm23R H23R Hr5Rx Hlow Hup).
L175983
Assume HxnotC5: ¬ (x ∈ C5).
L175983
We prove the intermediate claim HxX: x ∈ X.
L175985
An exact proof term for the current goal is (HAsubX x HxA).
L175985
We prove the intermediate claim HnotI1: ¬ (apply_fun r4s x ∈ I1).
L175987
Assume Hr4sI1': apply_fun r4s x ∈ I1.
L175987
We prove the intermediate claim Hr4sI1I: apply_fun r4s x ∈ I1 ∩ I.
L175989
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r4s x) Hr4sI1' Hr4sIx).
L175989
We prove the intermediate claim HxB5': x ∈ B5.
L175991
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r4s x0 ∈ I1 ∩ I) x HxA Hr4sI1I).
L175991
Apply FalseE to the current goal.
L175992
An exact proof term for the current goal is (HxnotB5 HxB5').
L175993
We prove the intermediate claim HnotI3: ¬ (apply_fun r4s x ∈ I3).
L175995
Assume Hr4sI3': apply_fun r4s x ∈ I3.
L175995
We prove the intermediate claim Hr4sI3I: apply_fun r4s x ∈ I3 ∩ I.
L175997
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r4s x) Hr4sI3' Hr4sIx).
L175997
We prove the intermediate claim HxC5': x ∈ C5.
L175999
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r4s x0 ∈ I3 ∩ I) x HxA Hr4sI3I).
L175999
Apply FalseE to the current goal.
L176000
An exact proof term for the current goal is (HxnotC5 HxC5').
L176001
We prove the intermediate claim Hr4sRx: apply_fun r4s x ∈ R.
L176003
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r4s x) Hr4sIx).
L176003
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176005
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176005
We prove the intermediate claim H13R: one_third ∈ R.
L176007
An exact proof term for the current goal is one_third_in_R.
L176007
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176009
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176009
We prove the intermediate claim Hr4s_boundsI: Rle (minus_SNo 1) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) 1.
L176011
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r4s x) Hm1R real_1 Hr4sIx).
L176011
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r4s x) (minus_SNo 1)).
L176013
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r4s x) (andEL (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_boundsI)).
L176014
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r4s x)).
L176016
An exact proof term for the current goal is (RleE_nlt (apply_fun r4s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r4s x)) (Rle (apply_fun r4s x) 1) Hr4s_boundsI)).
L176017
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r4s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r4s x).
L176019
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r4s x) Hm1R Hm13R Hr4sRx HnotI1).
L176020
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r4s x).
L176022
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r4s x))) to the current goal.
L176023
Assume Hbad: Rlt (apply_fun r4s x) (minus_SNo 1).
L176023
Apply FalseE to the current goal.
L176024
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L176026
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r4s x).
L176026
An exact proof term for the current goal is Hok.
L176027
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r4s x) (minus_SNo one_third)).
L176029
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r4s x) Hm13lt_fx).
L176029
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r4s x) one_third ∨ Rlt 1 (apply_fun r4s x).
L176031
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r4s x) H13R real_1 Hr4sRx HnotI3).
L176032
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r4s x) one_third.
L176034
Apply (HnotI3_cases (Rlt (apply_fun r4s x) one_third)) to the current goal.
L176035
Assume Hok: Rlt (apply_fun r4s x) one_third.
L176035
An exact proof term for the current goal is Hok.
L176037
Assume Hbad: Rlt 1 (apply_fun r4s x).
L176037
Apply FalseE to the current goal.
L176038
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L176039
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r4s x)).
L176041
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r4s x) one_third Hfx_lt_13).
L176041
We prove the intermediate claim Hr4sI0: apply_fun r4s x ∈ I0.
L176043
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L176045
We prove the intermediate claim HxSep: apply_fun r4s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L176047
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r4s x) Hr4sRx (andI (¬ (Rlt (apply_fun r4s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r4s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L176051
rewrite the current goal using HI0_def (from left to right).
L176052
An exact proof term for the current goal is HxSep.
L176053
We prove the intermediate claim Hu5funI0: function_on u5 X I0.
L176055
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u5 Hu5contI0).
L176055
We prove the intermediate claim Hu5xI0: apply_fun u5 x ∈ I0.
L176057
An exact proof term for the current goal is (Hu5funI0 x HxX).
L176057
We prove the intermediate claim Hu5xR: apply_fun u5 x ∈ R.
L176059
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u5 x) Hu5xI0).
L176059
We prove the intermediate claim Hm_u5x_R: minus_SNo (apply_fun u5 x) ∈ R.
L176061
An exact proof term for the current goal is (real_minus_SNo (apply_fun u5 x) Hu5xR).
L176061
rewrite the current goal using (Hr5_apply x HxA) (from left to right).
L176062
We prove the intermediate claim Hr5xR: add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) ∈ R.
L176064
An exact proof term for the current goal is (real_add_SNo (apply_fun r4s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) (minus_SNo (apply_fun u5 x)) Hm_u5x_R).
L176066
We prove the intermediate claim H23R: two_thirds ∈ R.
L176068
An exact proof term for the current goal is two_thirds_in_R.
L176068
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176070
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176070
We prove the intermediate claim Hr4s_bounds0: Rle (minus_SNo one_third) (apply_fun r4s x) ∧ Rle (apply_fun r4s x) one_third.
L176072
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r4s x) Hm13R H13R Hr4sI0).
L176072
We prove the intermediate claim Hu5_bounds0: Rle (minus_SNo one_third) (apply_fun u5 x) ∧ Rle (apply_fun u5 x) one_third.
L176074
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u5 x) Hm13R H13R Hu5xI0).
L176074
We prove the intermediate claim Hm13_le_r4s: Rle (minus_SNo one_third) (apply_fun r4s x).
L176076
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r4s x)) (Rle (apply_fun r4s x) one_third) Hr4s_bounds0).
L176076
We prove the intermediate claim Hr4s_le_13: Rle (apply_fun r4s x) one_third.
L176078
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r4s x)) (Rle (apply_fun r4s x) one_third) Hr4s_bounds0).
L176078
We prove the intermediate claim Hm13_le_u5x: Rle (minus_SNo one_third) (apply_fun u5 x).
L176080
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u5 x)) (Rle (apply_fun u5 x) one_third) Hu5_bounds0).
L176080
We prove the intermediate claim Hu5x_le_13: Rle (apply_fun u5 x) one_third.
L176082
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u5 x)) (Rle (apply_fun u5 x) one_third) Hu5_bounds0).
L176082
We prove the intermediate claim Hm13_le_mu5: Rle (minus_SNo one_third) (minus_SNo (apply_fun u5 x)).
L176084
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u5 x) one_third Hu5x_le_13).
L176084
We prove the intermediate claim Hmu5_le_13: Rle (minus_SNo (apply_fun u5 x)) one_third.
L176086
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u5 x)) (minus_SNo (minus_SNo one_third)).
L176087
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u5 x) Hm13_le_u5x).
L176087
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L176088
An exact proof term for the current goal is Htmp.
L176089
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))).
L176092
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u5 x)) Hm13R Hm13R Hm_u5x_R Hm13_le_mu5).
L176093
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L176096
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) Hm_u5x_R Hm13_le_r4s).
L176100
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L176103
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) Hlow1 Hlow2).
L176106
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))).
L176108
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L176108
An exact proof term for the current goal is Hlow_tmp.
L176109
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) one_third).
L176112
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r4s x) (minus_SNo (apply_fun u5 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) Hm_u5x_R H13R Hmu5_le_13).
L176114
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r4s x) one_third) (add_SNo one_third one_third).
L176117
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r4s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r4s x) Hr4sI0) H13R H13R Hr4s_le_13).
L176119
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo one_third one_third).
L176122
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) (add_SNo (apply_fun r4s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L176125
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L176127
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) two_thirds.
L176129
rewrite the current goal using Hdef23 (from left to right) at position 1.
L176129
An exact proof term for the current goal is Hup_tmp.
L176130
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r4s x) (minus_SNo (apply_fun u5 x))) Hm23R H23R Hr5xR Hlow Hup).
L176133
Set r5s to be the term compose_fun A r5 (div_const_fun den).
L176134
We prove the intermediate claim Hr5s_cont: continuous_map A Ta I Ti r5s.
L176136
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L176137
An exact proof term for the current goal is R_standard_topology_is_topology.
L176137
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L176139
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L176139
We prove the intermediate claim Hr5s_contR: continuous_map A Ta R R_standard_topology r5s.
L176141
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r5 (div_const_fun den) Hr5_cont Hdivcont).
L176142
We prove the intermediate claim Hr5s_I: ∀x : set, x ∈ A → apply_fun r5s x ∈ I.
L176144
Let x be given.
L176144
Assume HxA: x ∈ A.
L176144
We prove the intermediate claim Hr5xI2: apply_fun r5 x ∈ I2.
L176146
An exact proof term for the current goal is (Hr5_range x HxA).
L176146
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L176148
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176148
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L176150
An exact proof term for the current goal is (real_minus_SNo den H23R).
L176150
We prove the intermediate claim Hr5xR: apply_fun r5 x ∈ R.
L176152
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r5 x) Hr5xI2).
L176152
We prove the intermediate claim Hr5xS: SNo (apply_fun r5 x).
L176154
An exact proof term for the current goal is (real_SNo (apply_fun r5 x) Hr5xR).
L176154
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r5 x) ∧ Rle (apply_fun r5 x) den.
L176156
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r5 x) HmdenR H23R Hr5xI2).
L176156
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r5 x).
L176158
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r5 x)) (Rle (apply_fun r5 x) den) Hbounds).
L176160
We prove the intermediate claim Hhi: Rle (apply_fun r5 x) den.
L176162
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r5 x)) (Rle (apply_fun r5 x) den) Hbounds).
L176164
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r5 x)).
L176166
An exact proof term for the current goal is (RleE_nlt (apply_fun r5 x) den Hhi).
L176166
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r5 x) (minus_SNo den)).
L176168
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r5 x) Hlo).
L176168
We prove the intermediate claim HyEq: apply_fun r5s x = div_SNo (apply_fun r5 x) den.
L176170
rewrite the current goal using (compose_fun_apply A r5 (div_const_fun den) x HxA) (from left to right).
L176170
rewrite the current goal using (div_const_fun_apply den (apply_fun r5 x) H23R Hr5xR) (from left to right).
Use reflexivity.
L176172
We prove the intermediate claim HyR: apply_fun r5s x ∈ R.
L176174
rewrite the current goal using HyEq (from left to right).
L176174
An exact proof term for the current goal is (real_div_SNo (apply_fun r5 x) Hr5xR den H23R).
L176175
We prove the intermediate claim HyS: SNo (apply_fun r5s x).
L176177
An exact proof term for the current goal is (real_SNo (apply_fun r5s x) HyR).
L176177
We prove the intermediate claim Hy_le_1: Rle (apply_fun r5s x) 1.
L176179
Apply (RleI (apply_fun r5s x) 1 HyR real_1) to the current goal.
L176179
We will prove ¬ (Rlt 1 (apply_fun r5s x)).
L176180
Assume H1lt: Rlt 1 (apply_fun r5s x).
L176181
We prove the intermediate claim H1lty: 1 < apply_fun r5s x.
L176183
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r5s x) H1lt).
L176183
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r5s x) den.
L176185
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r5s x) den SNo_1 HyS H23S H23pos H1lty).
L176185
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r5s x) den.
L176187
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L176187
An exact proof term for the current goal is HmulLt.
L176188
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r5s x) den = apply_fun r5 x.
L176190
rewrite the current goal using HyEq (from left to right) at position 1.
L176190
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r5 x) den Hr5xS H23S H23ne0).
L176191
We prove the intermediate claim Hden_lt_r5x: den < apply_fun r5 x.
L176193
rewrite the current goal using HmulEq (from right to left).
L176193
An exact proof term for the current goal is HmulLt'.
L176194
We prove the intermediate claim Hbad: Rlt den (apply_fun r5 x).
L176196
An exact proof term for the current goal is (RltI den (apply_fun r5 x) H23R Hr5xR Hden_lt_r5x).
L176196
An exact proof term for the current goal is (Hnlt_hi Hbad).
L176197
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r5s x).
L176199
Apply (RleI (minus_SNo 1) (apply_fun r5s x) Hm1R HyR) to the current goal.
L176199
We will prove ¬ (Rlt (apply_fun r5s x) (minus_SNo 1)).
L176200
Assume Hylt: Rlt (apply_fun r5s x) (minus_SNo 1).
L176201
We prove the intermediate claim Hylts: apply_fun r5s x < minus_SNo 1.
L176203
An exact proof term for the current goal is (RltE_lt (apply_fun r5s x) (minus_SNo 1) Hylt).
L176203
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r5s x) den < mul_SNo (minus_SNo 1) den.
L176205
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r5s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L176206
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r5s x) den = apply_fun r5 x.
L176208
rewrite the current goal using HyEq (from left to right) at position 1.
L176208
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r5 x) den Hr5xS H23S H23ne0).
L176209
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L176211
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L176211
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L176213
We prove the intermediate claim Hr5x_lt_mden: apply_fun r5 x < minus_SNo den.
L176215
rewrite the current goal using HmulEq (from right to left).
L176215
rewrite the current goal using HrhsEq (from right to left).
L176216
An exact proof term for the current goal is HmulLt.
L176217
We prove the intermediate claim Hbad: Rlt (apply_fun r5 x) (minus_SNo den).
L176219
An exact proof term for the current goal is (RltI (apply_fun r5 x) (minus_SNo den) Hr5xR HmdenR Hr5x_lt_mden).
L176219
An exact proof term for the current goal is (Hnlt_lo Hbad).
L176220
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r5s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L176222
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r5s I Hr5s_contR HIcR Hr5s_I).
L176223
We prove the intermediate claim Hex_u6: ∃u6 : set, continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third).
(*** ninth correction step scaffold: build u6 and g6 from residual r5s ***)
L176233
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r5s Hnorm HA Hr5s_cont).
L176233
Apply Hex_u6 to the current goal.
L176234
Let u6 be given.
L176235
Assume Hu6.
L176235
We prove the intermediate claim Hu6contI0: continuous_map X Tx I0 T0 u6.
L176237
We prove the intermediate claim Hu6AB: continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third).
L176241
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L176247
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6) (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) Hu6AB).
L176252
We prove the intermediate claim Hu6contR: continuous_map X Tx R R_standard_topology u6.
L176254
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L176255
We prove the intermediate claim HI0subR: I0 ⊆ R.
L176257
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L176257
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u6 Hu6contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L176265
Set den6 to be the term mul_SNo den5 den.
L176266
We prove the intermediate claim Hden6R: den6 ∈ R.
L176268
An exact proof term for the current goal is (real_mul_SNo den5 Hden5R den H23R).
L176268
We prove the intermediate claim Hden6pos: 0 < den6.
L176270
We prove the intermediate claim Hden5S: SNo den5.
L176271
An exact proof term for the current goal is (real_SNo den5 Hden5R).
L176271
An exact proof term for the current goal is (mul_SNo_pos_pos den5 den Hden5S H23S Hden5pos HdenPos).
L176272
Set u6s to be the term compose_fun X u6 (mul_const_fun den6).
L176273
We prove the intermediate claim Hu6s_cont: continuous_map X Tx R R_standard_topology u6s.
L176275
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u6 den6 HTx Hu6contR Hden6R Hden6pos).
L176275
Set g6 to be the term compose_fun X (pair_map X g5 u6s) add_fun_R.
L176276
We prove the intermediate claim Hg6cont: continuous_map X Tx R R_standard_topology g6.
L176278
An exact proof term for the current goal is (add_two_continuous_R X Tx g5 u6s HTx Hg5cont Hu6s_cont).
L176278
We prove the intermediate claim Hu6contA: continuous_map A Ta R R_standard_topology u6.
(*** tenth correction step scaffold: residual r6 on A and scaling r6s ***)
L176282
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u6 A HTx HAsubX Hu6contR).
L176282
Set u6neg to be the term compose_fun A u6 neg_fun.
L176283
We prove the intermediate claim Hu6neg_cont: continuous_map A Ta R R_standard_topology u6neg.
L176285
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u6 neg_fun Hu6contA Hnegcont).
L176286
We prove the intermediate claim Hr5s_contR: continuous_map A Ta R R_standard_topology r5s.
L176288
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r5s Hr5s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L176289
Set r6 to be the term compose_fun A (pair_map A r5s u6neg) add_fun_R.
L176290
We prove the intermediate claim Hr6_cont: continuous_map A Ta R R_standard_topology r6.
L176292
An exact proof term for the current goal is (add_two_continuous_R A Ta r5s u6neg HTa Hr5s_contR Hu6neg_cont).
L176292
We prove the intermediate claim Hr6_apply: ∀x : set, x ∈ A → apply_fun r6 x = add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x)).
L176295
Let x be given.
L176295
Assume HxA: x ∈ A.
L176295
We prove the intermediate claim Hpimg: apply_fun (pair_map A r5s u6neg) x ∈ setprod R R.
L176297
rewrite the current goal using (pair_map_apply A R R r5s u6neg x HxA) (from left to right).
L176297
We prove the intermediate claim Hr5sxI: apply_fun r5s x ∈ I.
L176299
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r5s Hr5s_cont x HxA).
L176299
We prove the intermediate claim Hr5sxR: apply_fun r5s x ∈ R.
L176301
An exact proof term for the current goal is (HIcR (apply_fun r5s x) Hr5sxI).
L176301
We prove the intermediate claim Hu6negRx: apply_fun u6neg x ∈ R.
L176303
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u6neg Hu6neg_cont x HxA).
L176303
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r5s x) (apply_fun u6neg x) Hr5sxR Hu6negRx).
L176304
rewrite the current goal using (compose_fun_apply A (pair_map A r5s u6neg) add_fun_R x HxA) (from left to right).
L176305
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r5s u6neg) x) Hpimg) (from left to right) at position 1.
L176306
rewrite the current goal using (pair_map_apply A R R r5s u6neg x HxA) (from left to right).
L176307
rewrite the current goal using (tuple_2_0_eq (apply_fun r5s x) (apply_fun u6neg x)) (from left to right).
L176308
rewrite the current goal using (tuple_2_1_eq (apply_fun r5s x) (apply_fun u6neg x)) (from left to right).
L176309
rewrite the current goal using (compose_fun_apply A u6 neg_fun x HxA) (from left to right) at position 1.
L176310
We prove the intermediate claim Hu6Rx: apply_fun u6 x ∈ R.
L176312
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u6 Hu6contA x HxA).
L176312
rewrite the current goal using (neg_fun_apply (apply_fun u6 x) Hu6Rx) (from left to right) at position 1.
Use reflexivity.
L176314
We prove the intermediate claim Hr6_range: ∀x : set, x ∈ A → apply_fun r6 x ∈ I2.
L176316
We prove the intermediate claim Hu6AB: continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third).
(*** show the next residual stays within [-2/3,2/3] on A ***)
L176321
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L176327
We prove the intermediate claim Hu6_on_B6: ∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third.
L176331
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u6) (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third) Hu6AB).
L176335
We prove the intermediate claim Hu6_on_C6: ∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third.
L176339
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u6 ∧ (∀x : set, x ∈ preimage_of A r5s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u6 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r5s ((closed_interval one_third 1) ∩ I) → apply_fun u6 x = one_third) Hu6).
L176345
Let x be given.
L176346
Assume HxA: x ∈ A.
L176346
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L176347
Set I3 to be the term closed_interval one_third 1.
L176348
Set B6 to be the term preimage_of A r5s (I1 ∩ I).
L176349
Set C6 to be the term preimage_of A r5s (I3 ∩ I).
L176350
We prove the intermediate claim Hr5sIx: apply_fun r5s x ∈ I.
L176352
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r5s Hr5s_cont x HxA).
L176352
We prove the intermediate claim HB6_cases: x ∈ B6 ∨ ¬ (x ∈ B6).
L176354
An exact proof term for the current goal is (xm (x ∈ B6)).
L176354
Apply (HB6_cases (apply_fun r6 x ∈ I2)) to the current goal.
L176356
Assume HxB6: x ∈ B6.
L176356
We prove the intermediate claim Hu6eq: apply_fun u6 x = minus_SNo one_third.
L176358
An exact proof term for the current goal is (Hu6_on_B6 x HxB6).
L176358
We prove the intermediate claim Hr6eq: apply_fun r6 x = add_SNo (apply_fun r5s x) one_third.
L176360
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L176360
rewrite the current goal using Hu6eq (from left to right) at position 1.
L176361
We prove the intermediate claim H13R: one_third ∈ R.
L176363
An exact proof term for the current goal is one_third_in_R.
L176363
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L176365
rewrite the current goal using Hr6eq (from left to right).
L176366
We prove the intermediate claim Hr5sI1I: apply_fun r5s x ∈ I1 ∩ I.
L176368
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r5s x0 ∈ I1 ∩ I) x HxB6).
L176368
We prove the intermediate claim Hr5sI1: apply_fun r5s x ∈ I1.
L176370
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r5s x) Hr5sI1I).
L176370
We prove the intermediate claim H13R: one_third ∈ R.
L176372
An exact proof term for the current goal is one_third_in_R.
L176372
We prove the intermediate claim H23R: two_thirds ∈ R.
L176374
An exact proof term for the current goal is two_thirds_in_R.
L176374
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176376
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176376
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176378
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176378
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176380
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176380
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L176382
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hr5sI1).
L176382
We prove the intermediate claim Hr5s_bounds: Rle (minus_SNo 1) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) (minus_SNo one_third).
L176384
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hm1R Hm13R Hr5sI1).
L176385
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r5s x).
L176387
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) (minus_SNo one_third)) Hr5s_bounds).
L176388
We prove the intermediate claim HhiI1: Rle (apply_fun r5s x) (minus_SNo one_third).
L176390
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) (minus_SNo one_third)) Hr5s_bounds).
L176391
We prove the intermediate claim Hr6Rx: add_SNo (apply_fun r5s x) one_third ∈ R.
L176393
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) Hr5sRx one_third H13R).
L176393
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r5s x) one_third).
L176395
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r5s x) one_third Hm1R Hr5sRx H13R Hm1le).
L176395
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) one_third).
L176397
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L176397
An exact proof term for the current goal is Hlow_tmp.
L176398
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r5s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L176400
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) (minus_SNo one_third) one_third Hr5sRx Hm13R H13R HhiI1).
L176400
We prove the intermediate claim H13S: SNo one_third.
L176402
An exact proof term for the current goal is (real_SNo one_third H13R).
L176402
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r5s x) one_third) 0.
L176404
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L176404
An exact proof term for the current goal is Hup0_tmp.
L176405
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L176407
An exact proof term for the current goal is Rle_0_two_thirds.
L176407
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) one_third) two_thirds.
L176409
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) one_third) 0 two_thirds Hup0 H0le23).
L176409
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) one_third) Hm23R H23R Hr6Rx Hlow Hup).
L176413
Assume HxnotB6: ¬ (x ∈ B6).
L176413
We prove the intermediate claim HC6_cases: x ∈ C6 ∨ ¬ (x ∈ C6).
L176415
An exact proof term for the current goal is (xm (x ∈ C6)).
L176415
Apply (HC6_cases (apply_fun r6 x ∈ I2)) to the current goal.
L176417
Assume HxC6: x ∈ C6.
L176417
We prove the intermediate claim Hu6eq: apply_fun u6 x = one_third.
L176419
An exact proof term for the current goal is (Hu6_on_C6 x HxC6).
L176419
We prove the intermediate claim Hr6eq: apply_fun r6 x = add_SNo (apply_fun r5s x) (minus_SNo one_third).
L176421
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L176421
rewrite the current goal using Hu6eq (from left to right) at position 1.
Use reflexivity.
L176423
rewrite the current goal using Hr6eq (from left to right).
L176424
We prove the intermediate claim Hr5sI3I: apply_fun r5s x ∈ I3 ∩ I.
L176426
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r5s x0 ∈ I3 ∩ I) x HxC6).
L176426
We prove the intermediate claim Hr5sI3: apply_fun r5s x ∈ I3.
L176428
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r5s x) Hr5sI3I).
L176428
We prove the intermediate claim H13R: one_third ∈ R.
L176430
An exact proof term for the current goal is one_third_in_R.
L176430
We prove the intermediate claim H23R: two_thirds ∈ R.
L176432
An exact proof term for the current goal is two_thirds_in_R.
L176432
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176434
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176434
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176436
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176436
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L176438
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r5s x) Hr5sI3).
L176438
We prove the intermediate claim Hr5s_bounds: Rle one_third (apply_fun r5s x) ∧ Rle (apply_fun r5s x) 1.
L176440
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r5s x) H13R real_1 Hr5sI3).
L176440
We prove the intermediate claim HloI3: Rle one_third (apply_fun r5s x).
L176442
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_bounds).
L176442
We prove the intermediate claim HhiI3: Rle (apply_fun r5s x) 1.
L176444
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_bounds).
L176444
We prove the intermediate claim Hr6Rx: add_SNo (apply_fun r5s x) (minus_SNo one_third) ∈ R.
L176446
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) Hr5sRx (minus_SNo one_third) Hm13R).
L176446
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L176449
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r5s x) (minus_SNo one_third) H13R Hr5sRx Hm13R HloI3).
L176450
We prove the intermediate claim H13S: SNo one_third.
L176452
An exact proof term for the current goal is (real_SNo one_third H13R).
L176452
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L176454
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L176454
An exact proof term for the current goal is H0le_tmp.
L176455
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L176457
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L176457
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) (minus_SNo one_third)).
L176459
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r5s x) (minus_SNo one_third)) Hm23le0 H0le).
L176461
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r5s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L176464
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) 1 (minus_SNo one_third) Hr5sRx real_1 Hm13R HhiI3).
L176465
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L176467
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L176467
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L176468
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) (minus_SNo one_third)) two_thirds.
L176470
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L176473
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) (minus_SNo one_third)) Hm23R H23R Hr6Rx Hlow Hup).
L176477
Assume HxnotC6: ¬ (x ∈ C6).
L176477
We prove the intermediate claim HxX: x ∈ X.
L176479
An exact proof term for the current goal is (HAsubX x HxA).
L176479
We prove the intermediate claim HnotI1: ¬ (apply_fun r5s x ∈ I1).
L176481
Assume Hr5sI1': apply_fun r5s x ∈ I1.
L176481
We prove the intermediate claim Hr5sI1I: apply_fun r5s x ∈ I1 ∩ I.
L176483
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r5s x) Hr5sI1' Hr5sIx).
L176483
We prove the intermediate claim HxB6': x ∈ B6.
L176485
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r5s x0 ∈ I1 ∩ I) x HxA Hr5sI1I).
L176485
Apply FalseE to the current goal.
L176486
An exact proof term for the current goal is (HxnotB6 HxB6').
L176487
We prove the intermediate claim HnotI3: ¬ (apply_fun r5s x ∈ I3).
L176489
Assume Hr5sI3': apply_fun r5s x ∈ I3.
L176489
We prove the intermediate claim Hr5sI3I: apply_fun r5s x ∈ I3 ∩ I.
L176491
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r5s x) Hr5sI3' Hr5sIx).
L176491
We prove the intermediate claim HxC6': x ∈ C6.
L176493
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r5s x0 ∈ I3 ∩ I) x HxA Hr5sI3I).
L176493
Apply FalseE to the current goal.
L176494
An exact proof term for the current goal is (HxnotC6 HxC6').
L176495
We prove the intermediate claim Hr5sRx: apply_fun r5s x ∈ R.
L176497
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r5s x) Hr5sIx).
L176497
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176499
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176499
We prove the intermediate claim H13R: one_third ∈ R.
L176501
An exact proof term for the current goal is one_third_in_R.
L176501
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176503
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176503
We prove the intermediate claim Hr5s_boundsI: Rle (minus_SNo 1) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) 1.
L176505
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r5s x) Hm1R real_1 Hr5sIx).
L176505
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r5s x) (minus_SNo 1)).
L176507
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r5s x) (andEL (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_boundsI)).
L176508
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r5s x)).
L176510
An exact proof term for the current goal is (RleE_nlt (apply_fun r5s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r5s x)) (Rle (apply_fun r5s x) 1) Hr5s_boundsI)).
L176511
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r5s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r5s x).
L176513
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r5s x) Hm1R Hm13R Hr5sRx HnotI1).
L176514
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r5s x).
L176516
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r5s x))) to the current goal.
L176517
Assume Hbad: Rlt (apply_fun r5s x) (minus_SNo 1).
L176517
Apply FalseE to the current goal.
L176518
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L176520
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r5s x).
L176520
An exact proof term for the current goal is Hok.
L176521
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r5s x) (minus_SNo one_third)).
L176523
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r5s x) Hm13lt_fx).
L176523
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r5s x) one_third ∨ Rlt 1 (apply_fun r5s x).
L176525
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r5s x) H13R real_1 Hr5sRx HnotI3).
L176526
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r5s x) one_third.
L176528
Apply (HnotI3_cases (Rlt (apply_fun r5s x) one_third)) to the current goal.
L176529
Assume Hok: Rlt (apply_fun r5s x) one_third.
L176529
An exact proof term for the current goal is Hok.
L176531
Assume Hbad: Rlt 1 (apply_fun r5s x).
L176531
Apply FalseE to the current goal.
L176532
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L176533
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r5s x)).
L176535
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r5s x) one_third Hfx_lt_13).
L176535
We prove the intermediate claim Hr5sI0: apply_fun r5s x ∈ I0.
L176537
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L176539
We prove the intermediate claim HxSep: apply_fun r5s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L176541
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r5s x) Hr5sRx (andI (¬ (Rlt (apply_fun r5s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r5s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L176545
rewrite the current goal using HI0_def (from left to right).
L176546
An exact proof term for the current goal is HxSep.
L176547
We prove the intermediate claim Hu6funI0: function_on u6 X I0.
L176549
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u6 Hu6contI0).
L176549
We prove the intermediate claim Hu6xI0: apply_fun u6 x ∈ I0.
L176551
An exact proof term for the current goal is (Hu6funI0 x HxX).
L176551
We prove the intermediate claim Hu6xR: apply_fun u6 x ∈ R.
L176553
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u6 x) Hu6xI0).
L176553
We prove the intermediate claim Hm_u6x_R: minus_SNo (apply_fun u6 x) ∈ R.
L176555
An exact proof term for the current goal is (real_minus_SNo (apply_fun u6 x) Hu6xR).
L176555
rewrite the current goal using (Hr6_apply x HxA) (from left to right).
L176556
We prove the intermediate claim Hr6xR: add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) ∈ R.
L176558
An exact proof term for the current goal is (real_add_SNo (apply_fun r5s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) (minus_SNo (apply_fun u6 x)) Hm_u6x_R).
L176560
We prove the intermediate claim H23R: two_thirds ∈ R.
L176562
An exact proof term for the current goal is two_thirds_in_R.
L176562
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176564
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176564
We prove the intermediate claim Hr5s_bounds0: Rle (minus_SNo one_third) (apply_fun r5s x) ∧ Rle (apply_fun r5s x) one_third.
L176566
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r5s x) Hm13R H13R Hr5sI0).
L176566
We prove the intermediate claim Hu6_bounds0: Rle (minus_SNo one_third) (apply_fun u6 x) ∧ Rle (apply_fun u6 x) one_third.
L176568
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u6 x) Hm13R H13R Hu6xI0).
L176568
We prove the intermediate claim Hm13_le_r5s: Rle (minus_SNo one_third) (apply_fun r5s x).
L176570
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r5s x)) (Rle (apply_fun r5s x) one_third) Hr5s_bounds0).
L176570
We prove the intermediate claim Hr5s_le_13: Rle (apply_fun r5s x) one_third.
L176572
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r5s x)) (Rle (apply_fun r5s x) one_third) Hr5s_bounds0).
L176572
We prove the intermediate claim Hm13_le_u6x: Rle (minus_SNo one_third) (apply_fun u6 x).
L176574
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u6 x)) (Rle (apply_fun u6 x) one_third) Hu6_bounds0).
L176574
We prove the intermediate claim Hu6x_le_13: Rle (apply_fun u6 x) one_third.
L176576
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u6 x)) (Rle (apply_fun u6 x) one_third) Hu6_bounds0).
L176576
We prove the intermediate claim Hm13_le_mu6: Rle (minus_SNo one_third) (minus_SNo (apply_fun u6 x)).
L176578
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u6 x) one_third Hu6x_le_13).
L176578
We prove the intermediate claim Hmu6_le_13: Rle (minus_SNo (apply_fun u6 x)) one_third.
L176580
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u6 x)) (minus_SNo (minus_SNo one_third)).
L176581
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u6 x) Hm13_le_u6x).
L176581
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L176582
An exact proof term for the current goal is Htmp.
L176583
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))).
L176586
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u6 x)) Hm13R Hm13R Hm_u6x_R Hm13_le_mu6).
L176587
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L176590
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) Hm_u6x_R Hm13_le_r5s).
L176594
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L176597
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) Hlow1 Hlow2).
L176600
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))).
L176602
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L176602
An exact proof term for the current goal is Hlow_tmp.
L176603
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) one_third).
L176606
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r5s x) (minus_SNo (apply_fun u6 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) Hm_u6x_R H13R Hmu6_le_13).
L176608
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r5s x) one_third) (add_SNo one_third one_third).
L176611
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r5s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r5s x) Hr5sI0) H13R H13R Hr5s_le_13).
L176613
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo one_third one_third).
L176616
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) (add_SNo (apply_fun r5s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L176619
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L176621
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) two_thirds.
L176623
rewrite the current goal using Hdef23 (from left to right) at position 1.
L176623
An exact proof term for the current goal is Hup_tmp.
L176624
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r5s x) (minus_SNo (apply_fun u6 x))) Hm23R H23R Hr6xR Hlow Hup).
L176627
Set r6s to be the term compose_fun A r6 (div_const_fun den).
L176628
We prove the intermediate claim Hr6s_cont: continuous_map A Ta I Ti r6s.
L176630
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L176631
An exact proof term for the current goal is R_standard_topology_is_topology.
L176631
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L176633
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L176633
We prove the intermediate claim Hr6s_contR: continuous_map A Ta R R_standard_topology r6s.
L176635
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r6 (div_const_fun den) Hr6_cont Hdivcont).
L176636
We prove the intermediate claim Hr6s_I: ∀x : set, x ∈ A → apply_fun r6s x ∈ I.
L176638
Let x be given.
L176638
Assume HxA: x ∈ A.
L176638
We prove the intermediate claim Hr6xI2: apply_fun r6 x ∈ I2.
L176640
An exact proof term for the current goal is (Hr6_range x HxA).
L176640
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L176642
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176642
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L176644
An exact proof term for the current goal is (real_minus_SNo den H23R).
L176644
We prove the intermediate claim Hr6xR: apply_fun r6 x ∈ R.
L176646
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r6 x) Hr6xI2).
L176646
We prove the intermediate claim Hr6xS: SNo (apply_fun r6 x).
L176648
An exact proof term for the current goal is (real_SNo (apply_fun r6 x) Hr6xR).
L176648
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r6 x) ∧ Rle (apply_fun r6 x) den.
L176650
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r6 x) HmdenR H23R Hr6xI2).
L176650
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r6 x).
L176652
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r6 x)) (Rle (apply_fun r6 x) den) Hbounds).
L176654
We prove the intermediate claim Hhi: Rle (apply_fun r6 x) den.
L176656
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r6 x)) (Rle (apply_fun r6 x) den) Hbounds).
L176658
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r6 x)).
L176660
An exact proof term for the current goal is (RleE_nlt (apply_fun r6 x) den Hhi).
L176660
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r6 x) (minus_SNo den)).
L176662
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r6 x) Hlo).
L176662
We prove the intermediate claim HyEq: apply_fun r6s x = div_SNo (apply_fun r6 x) den.
L176664
rewrite the current goal using (compose_fun_apply A r6 (div_const_fun den) x HxA) (from left to right).
L176664
rewrite the current goal using (div_const_fun_apply den (apply_fun r6 x) H23R Hr6xR) (from left to right).
Use reflexivity.
L176666
We prove the intermediate claim HyR: apply_fun r6s x ∈ R.
L176668
rewrite the current goal using HyEq (from left to right).
L176668
An exact proof term for the current goal is (real_div_SNo (apply_fun r6 x) Hr6xR den H23R).
L176669
We prove the intermediate claim HyS: SNo (apply_fun r6s x).
L176671
An exact proof term for the current goal is (real_SNo (apply_fun r6s x) HyR).
L176671
We prove the intermediate claim Hy_le_1: Rle (apply_fun r6s x) 1.
L176673
Apply (RleI (apply_fun r6s x) 1 HyR real_1) to the current goal.
L176673
We will prove ¬ (Rlt 1 (apply_fun r6s x)).
L176674
Assume H1lt: Rlt 1 (apply_fun r6s x).
L176675
We prove the intermediate claim H1lty: 1 < apply_fun r6s x.
L176677
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r6s x) H1lt).
L176677
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r6s x) den.
L176679
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r6s x) den SNo_1 HyS H23S H23pos H1lty).
L176679
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r6s x) den.
L176681
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L176681
An exact proof term for the current goal is HmulLt.
L176682
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r6s x) den = apply_fun r6 x.
L176684
rewrite the current goal using HyEq (from left to right) at position 1.
L176684
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r6 x) den Hr6xS H23S H23ne0).
L176685
We prove the intermediate claim Hden_lt_r6x: den < apply_fun r6 x.
L176687
rewrite the current goal using HmulEq (from right to left).
L176687
An exact proof term for the current goal is HmulLt'.
L176688
We prove the intermediate claim Hbad: Rlt den (apply_fun r6 x).
L176690
An exact proof term for the current goal is (RltI den (apply_fun r6 x) H23R Hr6xR Hden_lt_r6x).
L176690
An exact proof term for the current goal is (Hnlt_hi Hbad).
L176691
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r6s x).
L176693
Apply (RleI (minus_SNo 1) (apply_fun r6s x) Hm1R HyR) to the current goal.
L176693
We will prove ¬ (Rlt (apply_fun r6s x) (minus_SNo 1)).
L176694
Assume Hylt: Rlt (apply_fun r6s x) (minus_SNo 1).
L176695
We prove the intermediate claim Hylts: apply_fun r6s x < minus_SNo 1.
L176697
An exact proof term for the current goal is (RltE_lt (apply_fun r6s x) (minus_SNo 1) Hylt).
L176697
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r6s x) den < mul_SNo (minus_SNo 1) den.
L176699
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r6s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L176700
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r6s x) den = apply_fun r6 x.
L176702
rewrite the current goal using HyEq (from left to right) at position 1.
L176702
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r6 x) den Hr6xS H23S H23ne0).
L176703
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L176705
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L176705
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L176707
We prove the intermediate claim Hr6x_lt_mden: apply_fun r6 x < minus_SNo den.
L176709
rewrite the current goal using HmulEq (from right to left).
L176709
rewrite the current goal using HrhsEq (from right to left).
L176710
An exact proof term for the current goal is HmulLt.
L176711
We prove the intermediate claim Hbad: Rlt (apply_fun r6 x) (minus_SNo den).
L176713
An exact proof term for the current goal is (RltI (apply_fun r6 x) (minus_SNo den) Hr6xR HmdenR Hr6x_lt_mden).
L176713
An exact proof term for the current goal is (Hnlt_lo Hbad).
L176714
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r6s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L176716
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r6s I Hr6s_contR HIcR Hr6s_I).
L176717
We prove the intermediate claim Hex_u7: ∃u7 : set, continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third).
(*** eleventh correction step scaffold: build u7 and g7 from residual r6s ***)
L176727
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r6s Hnorm HA Hr6s_cont).
L176727
Apply Hex_u7 to the current goal.
L176728
Let u7 be given.
L176729
Assume Hu7.
L176729
We prove the intermediate claim Hu7contI0: continuous_map X Tx I0 T0 u7.
L176731
We prove the intermediate claim Hu7AB: continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third).
L176735
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L176741
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7) (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) Hu7AB).
L176746
We prove the intermediate claim Hu7contR: continuous_map X Tx R R_standard_topology u7.
L176748
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L176749
We prove the intermediate claim HI0subR: I0 ⊆ R.
L176751
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L176751
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u7 Hu7contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L176759
Set den7 to be the term mul_SNo den6 den.
L176760
We prove the intermediate claim Hden7R: den7 ∈ R.
L176762
An exact proof term for the current goal is (real_mul_SNo den6 Hden6R den H23R).
L176762
We prove the intermediate claim Hden7pos: 0 < den7.
L176764
We prove the intermediate claim Hden6S: SNo den6.
L176765
An exact proof term for the current goal is (real_SNo den6 Hden6R).
L176765
An exact proof term for the current goal is (mul_SNo_pos_pos den6 den Hden6S H23S Hden6pos HdenPos).
L176766
Set u7s to be the term compose_fun X u7 (mul_const_fun den7).
L176767
We prove the intermediate claim Hu7s_cont: continuous_map X Tx R R_standard_topology u7s.
L176769
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u7 den7 HTx Hu7contR Hden7R Hden7pos).
L176769
Set g7 to be the term compose_fun X (pair_map X g6 u7s) add_fun_R.
L176770
We prove the intermediate claim Hg7cont: continuous_map X Tx R R_standard_topology g7.
L176772
An exact proof term for the current goal is (add_two_continuous_R X Tx g6 u7s HTx Hg6cont Hu7s_cont).
L176772
We prove the intermediate claim Hu7contA: continuous_map A Ta R R_standard_topology u7.
(*** twelfth correction step scaffold: residual r7 on A and scaling r7s ***)
L176776
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u7 A HTx HAsubX Hu7contR).
L176776
Set u7neg to be the term compose_fun A u7 neg_fun.
L176777
We prove the intermediate claim Hu7neg_cont: continuous_map A Ta R R_standard_topology u7neg.
L176779
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u7 neg_fun Hu7contA Hnegcont).
L176780
We prove the intermediate claim Hr6s_contR: continuous_map A Ta R R_standard_topology r6s.
L176782
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r6s Hr6s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L176783
Set r7 to be the term compose_fun A (pair_map A r6s u7neg) add_fun_R.
L176784
We prove the intermediate claim Hr7_cont: continuous_map A Ta R R_standard_topology r7.
L176786
An exact proof term for the current goal is (add_two_continuous_R A Ta r6s u7neg HTa Hr6s_contR Hu7neg_cont).
L176786
We prove the intermediate claim Hr7_apply: ∀x : set, x ∈ A → apply_fun r7 x = add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x)).
L176789
Let x be given.
L176789
Assume HxA: x ∈ A.
L176789
We prove the intermediate claim Hpimg: apply_fun (pair_map A r6s u7neg) x ∈ setprod R R.
L176791
rewrite the current goal using (pair_map_apply A R R r6s u7neg x HxA) (from left to right).
L176791
We prove the intermediate claim Hr6sxI: apply_fun r6s x ∈ I.
L176793
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r6s Hr6s_cont x HxA).
L176793
We prove the intermediate claim Hr6sxR: apply_fun r6s x ∈ R.
L176795
An exact proof term for the current goal is (HIcR (apply_fun r6s x) Hr6sxI).
L176795
We prove the intermediate claim Hu7negRx: apply_fun u7neg x ∈ R.
L176797
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u7neg Hu7neg_cont x HxA).
L176797
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r6s x) (apply_fun u7neg x) Hr6sxR Hu7negRx).
L176798
rewrite the current goal using (compose_fun_apply A (pair_map A r6s u7neg) add_fun_R x HxA) (from left to right).
L176799
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r6s u7neg) x) Hpimg) (from left to right) at position 1.
L176800
rewrite the current goal using (pair_map_apply A R R r6s u7neg x HxA) (from left to right).
L176801
rewrite the current goal using (tuple_2_0_eq (apply_fun r6s x) (apply_fun u7neg x)) (from left to right).
L176802
rewrite the current goal using (tuple_2_1_eq (apply_fun r6s x) (apply_fun u7neg x)) (from left to right).
L176803
rewrite the current goal using (compose_fun_apply A u7 neg_fun x HxA) (from left to right) at position 1.
L176804
We prove the intermediate claim Hu7Rx: apply_fun u7 x ∈ R.
L176806
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u7 Hu7contA x HxA).
L176806
rewrite the current goal using (neg_fun_apply (apply_fun u7 x) Hu7Rx) (from left to right) at position 1.
Use reflexivity.
L176808
We prove the intermediate claim Hr7_range: ∀x : set, x ∈ A → apply_fun r7 x ∈ I2.
L176810
We prove the intermediate claim Hu7AB: continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third).
L176814
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L176820
We prove the intermediate claim Hu7_on_B7: ∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third.
L176824
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u7) (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third) Hu7AB).
L176828
We prove the intermediate claim Hu7_on_C7: ∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third.
L176832
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u7 ∧ (∀x : set, x ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r6s ((closed_interval one_third 1) ∩ I) → apply_fun u7 x = one_third) Hu7).
L176838
Let x be given.
L176839
Assume HxA: x ∈ A.
L176839
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L176840
Set I3 to be the term closed_interval one_third 1.
L176841
Set B7 to be the term preimage_of A r6s (I1 ∩ I).
L176842
Set C7 to be the term preimage_of A r6s (I3 ∩ I).
L176843
We prove the intermediate claim Hr6sIx: apply_fun r6s x ∈ I.
L176845
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r6s Hr6s_cont x HxA).
L176845
We prove the intermediate claim HB7_cases: x ∈ B7 ∨ ¬ (x ∈ B7).
L176847
An exact proof term for the current goal is (xm (x ∈ B7)).
L176847
Apply (HB7_cases (apply_fun r7 x ∈ I2)) to the current goal.
L176849
Assume HxB7: x ∈ B7.
L176849
We prove the intermediate claim Hu7eq: apply_fun u7 x = minus_SNo one_third.
L176851
An exact proof term for the current goal is (Hu7_on_B7 x HxB7).
L176851
We prove the intermediate claim Hr7eq: apply_fun r7 x = add_SNo (apply_fun r6s x) one_third.
L176853
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L176853
rewrite the current goal using Hu7eq (from left to right) at position 1.
L176854
We prove the intermediate claim H13R: one_third ∈ R.
L176856
An exact proof term for the current goal is one_third_in_R.
L176856
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L176858
rewrite the current goal using Hr7eq (from left to right).
L176859
We prove the intermediate claim Hr6sI1I: apply_fun r6s x ∈ I1 ∩ I.
L176861
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r6s x0 ∈ I1 ∩ I) x HxB7).
L176861
We prove the intermediate claim Hr6sI1: apply_fun r6s x ∈ I1.
L176863
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r6s x) Hr6sI1I).
L176863
We prove the intermediate claim H13R: one_third ∈ R.
L176865
An exact proof term for the current goal is one_third_in_R.
L176865
We prove the intermediate claim H23R: two_thirds ∈ R.
L176867
An exact proof term for the current goal is two_thirds_in_R.
L176867
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176869
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176869
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176871
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176871
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176873
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176873
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L176875
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hr6sI1).
L176875
We prove the intermediate claim Hr6s_bounds: Rle (minus_SNo 1) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) (minus_SNo one_third).
L176877
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hm1R Hm13R Hr6sI1).
L176878
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r6s x).
L176880
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) (minus_SNo one_third)) Hr6s_bounds).
L176881
We prove the intermediate claim HhiI1: Rle (apply_fun r6s x) (minus_SNo one_third).
L176883
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) (minus_SNo one_third)) Hr6s_bounds).
L176884
We prove the intermediate claim Hr7Rx: add_SNo (apply_fun r6s x) one_third ∈ R.
L176886
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) Hr6sRx one_third H13R).
L176886
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r6s x) one_third).
L176888
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r6s x) one_third Hm1R Hr6sRx H13R Hm1le).
L176888
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) one_third).
L176890
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L176890
An exact proof term for the current goal is Hlow_tmp.
L176891
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r6s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L176893
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) (minus_SNo one_third) one_third Hr6sRx Hm13R H13R HhiI1).
L176893
We prove the intermediate claim H13S: SNo one_third.
L176895
An exact proof term for the current goal is (real_SNo one_third H13R).
L176895
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r6s x) one_third) 0.
L176897
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L176897
An exact proof term for the current goal is Hup0_tmp.
L176898
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L176900
An exact proof term for the current goal is Rle_0_two_thirds.
L176900
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) one_third) two_thirds.
L176902
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) one_third) 0 two_thirds Hup0 H0le23).
L176902
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) one_third) Hm23R H23R Hr7Rx Hlow Hup).
L176906
Assume HxnotB7: ¬ (x ∈ B7).
L176906
We prove the intermediate claim HC7_cases: x ∈ C7 ∨ ¬ (x ∈ C7).
L176908
An exact proof term for the current goal is (xm (x ∈ C7)).
L176908
Apply (HC7_cases (apply_fun r7 x ∈ I2)) to the current goal.
L176910
Assume HxC7: x ∈ C7.
L176910
We prove the intermediate claim Hu7eq: apply_fun u7 x = one_third.
L176912
An exact proof term for the current goal is (Hu7_on_C7 x HxC7).
L176912
We prove the intermediate claim Hr7eq: apply_fun r7 x = add_SNo (apply_fun r6s x) (minus_SNo one_third).
L176914
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L176914
rewrite the current goal using Hu7eq (from left to right) at position 1.
Use reflexivity.
L176916
rewrite the current goal using Hr7eq (from left to right).
L176917
We prove the intermediate claim Hr6sI3I: apply_fun r6s x ∈ I3 ∩ I.
L176919
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r6s x0 ∈ I3 ∩ I) x HxC7).
L176919
We prove the intermediate claim Hr6sI3: apply_fun r6s x ∈ I3.
L176921
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r6s x) Hr6sI3I).
L176921
We prove the intermediate claim H13R: one_third ∈ R.
L176923
An exact proof term for the current goal is one_third_in_R.
L176923
We prove the intermediate claim H23R: two_thirds ∈ R.
L176925
An exact proof term for the current goal is two_thirds_in_R.
L176925
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L176927
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L176927
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176929
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176929
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L176931
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r6s x) Hr6sI3).
L176931
We prove the intermediate claim Hr6s_bounds: Rle one_third (apply_fun r6s x) ∧ Rle (apply_fun r6s x) 1.
L176933
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r6s x) H13R real_1 Hr6sI3).
L176933
We prove the intermediate claim HloI3: Rle one_third (apply_fun r6s x).
L176935
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_bounds).
L176935
We prove the intermediate claim HhiI3: Rle (apply_fun r6s x) 1.
L176937
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_bounds).
L176937
We prove the intermediate claim Hr7Rx: add_SNo (apply_fun r6s x) (minus_SNo one_third) ∈ R.
L176939
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) Hr6sRx (minus_SNo one_third) Hm13R).
L176939
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L176942
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r6s x) (minus_SNo one_third) H13R Hr6sRx Hm13R HloI3).
L176943
We prove the intermediate claim H13S: SNo one_third.
L176945
An exact proof term for the current goal is (real_SNo one_third H13R).
L176945
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L176947
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L176947
An exact proof term for the current goal is H0le_tmp.
L176948
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L176950
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L176950
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) (minus_SNo one_third)).
L176952
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r6s x) (minus_SNo one_third)) Hm23le0 H0le).
L176954
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r6s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L176957
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) 1 (minus_SNo one_third) Hr6sRx real_1 Hm13R HhiI3).
L176958
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L176960
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L176960
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L176961
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) (minus_SNo one_third)) two_thirds.
L176963
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L176966
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) (minus_SNo one_third)) Hm23R H23R Hr7Rx Hlow Hup).
L176970
Assume HxnotC7: ¬ (x ∈ C7).
L176970
We prove the intermediate claim HxX: x ∈ X.
L176972
An exact proof term for the current goal is (HAsubX x HxA).
L176972
We prove the intermediate claim HnotI1: ¬ (apply_fun r6s x ∈ I1).
L176974
Assume Hr6sI1': apply_fun r6s x ∈ I1.
L176974
We prove the intermediate claim Hr6sI1I: apply_fun r6s x ∈ I1 ∩ I.
L176976
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r6s x) Hr6sI1' Hr6sIx).
L176976
We prove the intermediate claim HxB7': x ∈ B7.
L176978
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r6s x0 ∈ I1 ∩ I) x HxA Hr6sI1I).
L176978
Apply FalseE to the current goal.
L176979
An exact proof term for the current goal is (HxnotB7 HxB7').
L176980
We prove the intermediate claim HnotI3: ¬ (apply_fun r6s x ∈ I3).
L176982
Assume Hr6sI3': apply_fun r6s x ∈ I3.
L176982
We prove the intermediate claim Hr6sI3I: apply_fun r6s x ∈ I3 ∩ I.
L176984
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r6s x) Hr6sI3' Hr6sIx).
L176984
We prove the intermediate claim HxC7': x ∈ C7.
L176986
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r6s x0 ∈ I3 ∩ I) x HxA Hr6sI3I).
L176986
Apply FalseE to the current goal.
L176987
An exact proof term for the current goal is (HxnotC7 HxC7').
L176988
We prove the intermediate claim Hr6sRx: apply_fun r6s x ∈ R.
L176990
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r6s x) Hr6sIx).
L176990
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L176992
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L176992
We prove the intermediate claim H13R: one_third ∈ R.
L176994
An exact proof term for the current goal is one_third_in_R.
L176994
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L176996
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L176996
We prove the intermediate claim Hr6s_boundsI: Rle (minus_SNo 1) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) 1.
L176998
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r6s x) Hm1R real_1 Hr6sIx).
L176998
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r6s x) (minus_SNo 1)).
L177000
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r6s x) (andEL (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_boundsI)).
L177001
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r6s x)).
L177003
An exact proof term for the current goal is (RleE_nlt (apply_fun r6s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r6s x)) (Rle (apply_fun r6s x) 1) Hr6s_boundsI)).
L177004
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r6s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r6s x).
L177006
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r6s x) Hm1R Hm13R Hr6sRx HnotI1).
L177007
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r6s x).
L177009
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r6s x))) to the current goal.
L177010
Assume Hbad: Rlt (apply_fun r6s x) (minus_SNo 1).
L177010
Apply FalseE to the current goal.
L177011
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L177013
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r6s x).
L177013
An exact proof term for the current goal is Hok.
L177014
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r6s x) (minus_SNo one_third)).
L177016
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r6s x) Hm13lt_fx).
L177016
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r6s x) one_third ∨ Rlt 1 (apply_fun r6s x).
L177018
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r6s x) H13R real_1 Hr6sRx HnotI3).
L177019
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r6s x) one_third.
L177021
Apply (HnotI3_cases (Rlt (apply_fun r6s x) one_third)) to the current goal.
L177022
Assume Hok: Rlt (apply_fun r6s x) one_third.
L177022
An exact proof term for the current goal is Hok.
L177024
Assume Hbad: Rlt 1 (apply_fun r6s x).
L177024
Apply FalseE to the current goal.
L177025
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L177026
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r6s x)).
L177028
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r6s x) one_third Hfx_lt_13).
L177028
We prove the intermediate claim Hr6sI0: apply_fun r6s x ∈ I0.
L177030
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L177031
We prove the intermediate claim HxSep: apply_fun r6s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L177033
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r6s x) Hr6sRx (andI (¬ (Rlt (apply_fun r6s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r6s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L177037
rewrite the current goal using HI0_def (from left to right).
L177038
An exact proof term for the current goal is HxSep.
L177039
We prove the intermediate claim Hu7contI0: continuous_map X Tx I0 T0 u7.
L177041
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u7) (∀x0 : set, x0 ∈ preimage_of A r6s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u7 x0 = minus_SNo one_third) Hu7AB).
L177045
We prove the intermediate claim Hu7funI0: function_on u7 X I0.
L177047
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u7 Hu7contI0).
L177047
We prove the intermediate claim Hu7xI0: apply_fun u7 x ∈ I0.
L177049
An exact proof term for the current goal is (Hu7funI0 x HxX).
L177049
We prove the intermediate claim Hu7xR: apply_fun u7 x ∈ R.
L177051
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u7 x) Hu7xI0).
L177051
We prove the intermediate claim Hm_u7x_R: minus_SNo (apply_fun u7 x) ∈ R.
L177053
An exact proof term for the current goal is (real_minus_SNo (apply_fun u7 x) Hu7xR).
L177053
rewrite the current goal using (Hr7_apply x HxA) (from left to right).
L177054
We prove the intermediate claim Hr7xR: add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) ∈ R.
L177056
An exact proof term for the current goal is (real_add_SNo (apply_fun r6s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) (minus_SNo (apply_fun u7 x)) Hm_u7x_R).
L177058
We prove the intermediate claim H23R: two_thirds ∈ R.
L177060
An exact proof term for the current goal is two_thirds_in_R.
L177060
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177062
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177062
We prove the intermediate claim Hr6s_bounds0: Rle (minus_SNo one_third) (apply_fun r6s x) ∧ Rle (apply_fun r6s x) one_third.
L177064
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r6s x) Hm13R H13R Hr6sI0).
L177064
We prove the intermediate claim Hu7_bounds0: Rle (minus_SNo one_third) (apply_fun u7 x) ∧ Rle (apply_fun u7 x) one_third.
L177066
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u7 x) Hm13R H13R Hu7xI0).
L177066
We prove the intermediate claim Hm13_le_r6s: Rle (minus_SNo one_third) (apply_fun r6s x).
L177068
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r6s x)) (Rle (apply_fun r6s x) one_third) Hr6s_bounds0).
L177068
We prove the intermediate claim Hr6s_le_13: Rle (apply_fun r6s x) one_third.
L177070
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r6s x)) (Rle (apply_fun r6s x) one_third) Hr6s_bounds0).
L177070
We prove the intermediate claim Hm13_le_u7x: Rle (minus_SNo one_third) (apply_fun u7 x).
L177072
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u7 x)) (Rle (apply_fun u7 x) one_third) Hu7_bounds0).
L177072
We prove the intermediate claim Hu7x_le_13: Rle (apply_fun u7 x) one_third.
L177074
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u7 x)) (Rle (apply_fun u7 x) one_third) Hu7_bounds0).
L177074
We prove the intermediate claim Hm13_le_mu7: Rle (minus_SNo one_third) (minus_SNo (apply_fun u7 x)).
L177076
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u7 x) one_third Hu7x_le_13).
L177076
We prove the intermediate claim Hmu7_le_13: Rle (minus_SNo (apply_fun u7 x)) one_third.
L177078
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u7 x)) (minus_SNo (minus_SNo one_third)).
L177079
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u7 x) Hm13_le_u7x).
L177079
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L177080
An exact proof term for the current goal is Htmp.
L177081
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))).
L177084
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u7 x)) Hm13R Hm13R Hm_u7x_R Hm13_le_mu7).
L177085
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L177088
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) Hm_u7x_R Hm13_le_r6s).
L177092
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L177095
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) Hlow1 Hlow2).
L177098
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))).
L177100
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L177100
An exact proof term for the current goal is Hlow_tmp.
L177101
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) one_third).
L177104
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r6s x) (minus_SNo (apply_fun u7 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) Hm_u7x_R H13R Hmu7_le_13).
L177106
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r6s x) one_third) (add_SNo one_third one_third).
L177109
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r6s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r6s x) Hr6sI0) H13R H13R Hr6s_le_13).
L177111
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo one_third one_third).
L177114
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) (add_SNo (apply_fun r6s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L177117
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L177119
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) two_thirds.
L177121
rewrite the current goal using Hdef23 (from left to right) at position 1.
L177121
An exact proof term for the current goal is Hup_tmp.
L177122
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r6s x) (minus_SNo (apply_fun u7 x))) Hm23R H23R Hr7xR Hlow Hup).
L177125
Set r7s to be the term compose_fun A r7 (div_const_fun den).
L177126
We prove the intermediate claim Hr7s_cont: continuous_map A Ta I Ti r7s.
L177128
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L177129
An exact proof term for the current goal is R_standard_topology_is_topology.
L177129
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L177131
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L177131
We prove the intermediate claim Hr7s_contR: continuous_map A Ta R R_standard_topology r7s.
L177133
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r7 (div_const_fun den) Hr7_cont Hdivcont).
L177134
We prove the intermediate claim Hr7s_I: ∀x : set, x ∈ A → apply_fun r7s x ∈ I.
L177136
Let x be given.
L177136
Assume HxA: x ∈ A.
L177136
We prove the intermediate claim Hr7xI2: apply_fun r7 x ∈ I2.
L177138
An exact proof term for the current goal is (Hr7_range x HxA).
L177138
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L177140
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177140
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L177142
An exact proof term for the current goal is (real_minus_SNo den H23R).
L177142
We prove the intermediate claim Hr7xR: apply_fun r7 x ∈ R.
L177144
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r7 x) Hr7xI2).
L177144
We prove the intermediate claim Hr7xS: SNo (apply_fun r7 x).
L177146
An exact proof term for the current goal is (real_SNo (apply_fun r7 x) Hr7xR).
L177146
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r7 x) ∧ Rle (apply_fun r7 x) den.
L177148
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r7 x) HmdenR H23R Hr7xI2).
L177148
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r7 x).
L177150
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r7 x)) (Rle (apply_fun r7 x) den) Hbounds).
L177152
We prove the intermediate claim Hhi: Rle (apply_fun r7 x) den.
L177154
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r7 x)) (Rle (apply_fun r7 x) den) Hbounds).
L177156
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r7 x)).
L177158
An exact proof term for the current goal is (RleE_nlt (apply_fun r7 x) den Hhi).
L177158
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r7 x) (minus_SNo den)).
L177160
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r7 x) Hlo).
L177160
We prove the intermediate claim HyEq: apply_fun r7s x = div_SNo (apply_fun r7 x) den.
L177162
rewrite the current goal using (compose_fun_apply A r7 (div_const_fun den) x HxA) (from left to right).
L177162
rewrite the current goal using (div_const_fun_apply den (apply_fun r7 x) H23R Hr7xR) (from left to right).
Use reflexivity.
L177164
We prove the intermediate claim HyR: apply_fun r7s x ∈ R.
L177166
rewrite the current goal using HyEq (from left to right).
L177166
An exact proof term for the current goal is (real_div_SNo (apply_fun r7 x) Hr7xR den H23R).
L177167
We prove the intermediate claim HyS: SNo (apply_fun r7s x).
L177169
An exact proof term for the current goal is (real_SNo (apply_fun r7s x) HyR).
L177169
We prove the intermediate claim Hy_le_1: Rle (apply_fun r7s x) 1.
L177171
Apply (RleI (apply_fun r7s x) 1 HyR real_1) to the current goal.
L177171
We will prove ¬ (Rlt 1 (apply_fun r7s x)).
L177172
Assume H1lt: Rlt 1 (apply_fun r7s x).
L177173
We prove the intermediate claim H1lty: 1 < apply_fun r7s x.
L177175
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r7s x) H1lt).
L177175
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r7s x) den.
L177177
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r7s x) den SNo_1 HyS H23S H23pos H1lty).
L177177
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r7s x) den.
L177179
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L177179
An exact proof term for the current goal is HmulLt.
L177180
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r7s x) den = apply_fun r7 x.
L177182
rewrite the current goal using HyEq (from left to right) at position 1.
L177182
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r7 x) den Hr7xS H23S H23ne0).
L177183
We prove the intermediate claim Hden_lt_r7x: den < apply_fun r7 x.
L177185
rewrite the current goal using HmulEq (from right to left).
L177185
An exact proof term for the current goal is HmulLt'.
L177186
We prove the intermediate claim Hbad: Rlt den (apply_fun r7 x).
L177188
An exact proof term for the current goal is (RltI den (apply_fun r7 x) H23R Hr7xR Hden_lt_r7x).
L177188
An exact proof term for the current goal is (Hnlt_hi Hbad).
L177189
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r7s x).
L177191
Apply (RleI (minus_SNo 1) (apply_fun r7s x) Hm1R HyR) to the current goal.
L177191
We will prove ¬ (Rlt (apply_fun r7s x) (minus_SNo 1)).
L177192
Assume Hylt: Rlt (apply_fun r7s x) (minus_SNo 1).
L177193
We prove the intermediate claim Hylts: apply_fun r7s x < minus_SNo 1.
L177195
An exact proof term for the current goal is (RltE_lt (apply_fun r7s x) (minus_SNo 1) Hylt).
L177195
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r7s x) den < mul_SNo (minus_SNo 1) den.
L177197
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r7s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L177198
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r7s x) den = apply_fun r7 x.
L177200
rewrite the current goal using HyEq (from left to right) at position 1.
L177200
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r7 x) den Hr7xS H23S H23ne0).
L177201
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L177203
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L177203
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L177205
We prove the intermediate claim Hr7x_lt_mden: apply_fun r7 x < minus_SNo den.
L177207
rewrite the current goal using HmulEq (from right to left).
L177207
rewrite the current goal using HrhsEq (from right to left).
L177208
An exact proof term for the current goal is HmulLt.
L177209
We prove the intermediate claim Hbad: Rlt (apply_fun r7 x) (minus_SNo den).
L177211
An exact proof term for the current goal is (RltI (apply_fun r7 x) (minus_SNo den) Hr7xR HmdenR Hr7x_lt_mden).
L177211
An exact proof term for the current goal is (Hnlt_lo Hbad).
L177212
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r7s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L177214
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r7s I Hr7s_contR HIcR Hr7s_I).
L177215
We prove the intermediate claim Hex_u8: ∃u8 : set, continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x = one_third).
(*** thirteenth correction step scaffold: build u8 and g8 from residual r7s ***)
L177225
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r7s Hnorm HA Hr7s_cont).
L177225
Apply Hex_u8 to the current goal.
L177226
Let u8 be given.
L177227
Assume Hu8.
L177227
We prove the intermediate claim Hu8contI0: continuous_map X Tx I0 T0 u8.
L177229
We prove the intermediate claim Hu8AB: continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third).
L177233
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8 ∧ (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x = one_third) Hu8).
L177239
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8) (∀x : set, x ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x = minus_SNo one_third) Hu8AB).
L177244
We prove the intermediate claim Hu8contR: continuous_map X Tx R R_standard_topology u8.
L177246
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L177247
We prove the intermediate claim HI0subR: I0 ⊆ R.
L177249
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L177249
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u8 Hu8contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L177257
Set den8 to be the term mul_SNo den7 den.
L177258
We prove the intermediate claim Hden8R: den8 ∈ R.
L177260
An exact proof term for the current goal is (real_mul_SNo den7 Hden7R den H23R).
L177260
We prove the intermediate claim Hden8pos: 0 < den8.
L177262
We prove the intermediate claim Hden7S: SNo den7.
L177263
An exact proof term for the current goal is (real_SNo den7 Hden7R).
L177263
An exact proof term for the current goal is (mul_SNo_pos_pos den7 den Hden7S H23S Hden7pos HdenPos).
L177264
Set u8s to be the term compose_fun X u8 (mul_const_fun den8).
L177265
We prove the intermediate claim Hu8s_cont: continuous_map X Tx R R_standard_topology u8s.
L177267
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u8 den8 HTx Hu8contR Hden8R Hden8pos).
L177267
Set g8 to be the term compose_fun X (pair_map X g7 u8s) add_fun_R.
L177268
We prove the intermediate claim Hg8cont: continuous_map X Tx R R_standard_topology g8.
L177270
An exact proof term for the current goal is (add_two_continuous_R X Tx g7 u8s HTx Hg7cont Hu8s_cont).
L177270
We prove the intermediate claim Hu8contA: continuous_map A Ta R R_standard_topology u8.
(*** fourteenth correction step scaffold: residual r8 on A and scaling r8s ***)
L177274
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u8 A HTx HAsubX Hu8contR).
L177274
Set u8neg to be the term compose_fun A u8 neg_fun.
L177275
We prove the intermediate claim Hu8neg_cont: continuous_map A Ta R R_standard_topology u8neg.
L177277
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u8 neg_fun Hu8contA Hnegcont).
L177278
We prove the intermediate claim Hr7s_contR: continuous_map A Ta R R_standard_topology r7s.
L177280
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r7s Hr7s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L177281
Set r8 to be the term compose_fun A (pair_map A r7s u8neg) add_fun_R.
L177282
We prove the intermediate claim Hr8_cont: continuous_map A Ta R R_standard_topology r8.
L177284
An exact proof term for the current goal is (add_two_continuous_R A Ta r7s u8neg HTa Hr7s_contR Hu8neg_cont).
L177284
We prove the intermediate claim Hr8_apply: ∀x : set, x ∈ A → apply_fun r8 x = add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x)).
L177287
Let x be given.
L177287
Assume HxA: x ∈ A.
L177287
We prove the intermediate claim Hpimg: apply_fun (pair_map A r7s u8neg) x ∈ setprod R R.
L177289
rewrite the current goal using (pair_map_apply A R R r7s u8neg x HxA) (from left to right).
L177289
We prove the intermediate claim Hr7sxI: apply_fun r7s x ∈ I.
L177291
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r7s Hr7s_cont x HxA).
L177291
We prove the intermediate claim Hr7sxR: apply_fun r7s x ∈ R.
L177293
An exact proof term for the current goal is (HIcR (apply_fun r7s x) Hr7sxI).
L177293
We prove the intermediate claim Hu8negRx: apply_fun u8neg x ∈ R.
L177295
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u8neg Hu8neg_cont x HxA).
L177295
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r7s x) (apply_fun u8neg x) Hr7sxR Hu8negRx).
L177296
rewrite the current goal using (compose_fun_apply A (pair_map A r7s u8neg) add_fun_R x HxA) (from left to right).
L177297
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r7s u8neg) x) Hpimg) (from left to right) at position 1.
L177298
rewrite the current goal using (pair_map_apply A R R r7s u8neg x HxA) (from left to right).
L177299
rewrite the current goal using (tuple_2_0_eq (apply_fun r7s x) (apply_fun u8neg x)) (from left to right).
L177300
rewrite the current goal using (tuple_2_1_eq (apply_fun r7s x) (apply_fun u8neg x)) (from left to right).
L177301
rewrite the current goal using (compose_fun_apply A u8 neg_fun x HxA) (from left to right) at position 1.
L177302
We prove the intermediate claim Hu8Rx: apply_fun u8 x ∈ R.
L177304
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u8 Hu8contA x HxA).
L177304
rewrite the current goal using (neg_fun_apply (apply_fun u8 x) Hu8Rx) (from left to right) at position 1.
Use reflexivity.
L177306
We prove the intermediate claim Hr8_range: ∀x : set, x ∈ A → apply_fun r8 x ∈ I2.
L177308
We prove the intermediate claim Hu8AB: continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third).
L177312
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third) Hu8).
L177318
We prove the intermediate claim Hu8_on_B8: ∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third.
L177322
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u8) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third) Hu8AB).
L177326
We prove the intermediate claim Hu8_on_C8: ∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third.
L177330
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u8 ∧ (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u8 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r7s ((closed_interval one_third 1) ∩ I) → apply_fun u8 x0 = one_third) Hu8).
L177336
Let x be given.
L177337
Assume HxA: x ∈ A.
L177337
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L177338
Set I3 to be the term closed_interval one_third 1.
L177339
Set B8 to be the term preimage_of A r7s (I1 ∩ I).
L177340
Set C8 to be the term preimage_of A r7s (I3 ∩ I).
L177341
We prove the intermediate claim Hr7sIx: apply_fun r7s x ∈ I.
L177343
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r7s Hr7s_cont x HxA).
L177343
We prove the intermediate claim HB8_cases: x ∈ B8 ∨ ¬ (x ∈ B8).
L177345
An exact proof term for the current goal is (xm (x ∈ B8)).
L177345
Apply (HB8_cases (apply_fun r8 x ∈ I2)) to the current goal.
L177347
Assume HxB8: x ∈ B8.
L177347
We prove the intermediate claim Hu8eq: apply_fun u8 x = minus_SNo one_third.
L177349
An exact proof term for the current goal is (Hu8_on_B8 x HxB8).
L177349
We prove the intermediate claim Hr8eq: apply_fun r8 x = add_SNo (apply_fun r7s x) one_third.
L177351
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L177351
rewrite the current goal using Hu8eq (from left to right) at position 1.
L177352
We prove the intermediate claim H13R: one_third ∈ R.
L177354
An exact proof term for the current goal is one_third_in_R.
L177354
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L177356
rewrite the current goal using Hr8eq (from left to right).
L177357
We prove the intermediate claim Hr7sI1I: apply_fun r7s x ∈ I1 ∩ I.
L177359
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r7s x0 ∈ I1 ∩ I) x HxB8).
L177359
We prove the intermediate claim Hr7sI1: apply_fun r7s x ∈ I1.
L177361
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r7s x) Hr7sI1I).
L177361
We prove the intermediate claim H13R: one_third ∈ R.
L177363
An exact proof term for the current goal is one_third_in_R.
L177363
We prove the intermediate claim H23R: two_thirds ∈ R.
L177365
An exact proof term for the current goal is two_thirds_in_R.
L177365
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177367
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177367
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177369
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177369
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177371
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177371
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L177373
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hr7sI1).
L177373
We prove the intermediate claim Hr7s_bounds: Rle (minus_SNo 1) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) (minus_SNo one_third).
L177375
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hm1R Hm13R Hr7sI1).
L177376
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r7s x).
L177378
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) (minus_SNo one_third)) Hr7s_bounds).
L177379
We prove the intermediate claim HhiI1: Rle (apply_fun r7s x) (minus_SNo one_third).
L177381
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) (minus_SNo one_third)) Hr7s_bounds).
L177382
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) one_third ∈ R.
L177384
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) Hr7sRx one_third H13R).
L177384
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r7s x) one_third).
L177386
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r7s x) one_third Hm1R Hr7sRx H13R Hm1le).
L177386
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) one_third).
L177388
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L177388
An exact proof term for the current goal is Hlow_tmp.
L177389
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r7s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L177391
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) (minus_SNo one_third) one_third Hr7sRx Hm13R H13R HhiI1).
L177391
We prove the intermediate claim H13S: SNo one_third.
L177393
An exact proof term for the current goal is (real_SNo one_third H13R).
L177393
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r7s x) one_third) 0.
L177395
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L177395
An exact proof term for the current goal is Hup0_tmp.
L177396
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L177398
An exact proof term for the current goal is Rle_0_two_thirds.
L177398
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) one_third) two_thirds.
L177400
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) one_third) 0 two_thirds Hup0 H0le23).
L177400
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) one_third) Hm23R H23R Hr8xR Hlow Hup).
L177404
Assume HxnotB8: ¬ (x ∈ B8).
L177404
We prove the intermediate claim HC8_cases: x ∈ C8 ∨ ¬ (x ∈ C8).
L177406
An exact proof term for the current goal is (xm (x ∈ C8)).
L177406
Apply (HC8_cases (apply_fun r8 x ∈ I2)) to the current goal.
L177408
Assume HxC8: x ∈ C8.
L177408
We prove the intermediate claim Hu8eq: apply_fun u8 x = one_third.
L177410
An exact proof term for the current goal is (Hu8_on_C8 x HxC8).
L177410
We prove the intermediate claim Hr8eq: apply_fun r8 x = add_SNo (apply_fun r7s x) (minus_SNo one_third).
L177412
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L177412
rewrite the current goal using Hu8eq (from left to right) at position 1.
Use reflexivity.
L177414
rewrite the current goal using Hr8eq (from left to right).
L177415
We prove the intermediate claim Hr7sI3I: apply_fun r7s x ∈ I3 ∩ I.
L177417
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r7s x0 ∈ I3 ∩ I) x HxC8).
L177417
We prove the intermediate claim Hr7sI3: apply_fun r7s x ∈ I3.
L177419
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r7s x) Hr7sI3I).
L177419
We prove the intermediate claim H13R: one_third ∈ R.
L177421
An exact proof term for the current goal is one_third_in_R.
L177421
We prove the intermediate claim H23R: two_thirds ∈ R.
L177423
An exact proof term for the current goal is two_thirds_in_R.
L177423
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177425
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177425
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177427
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177427
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L177429
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r7s x) Hr7sI3).
L177429
We prove the intermediate claim Hr7s_bounds: Rle one_third (apply_fun r7s x) ∧ Rle (apply_fun r7s x) 1.
L177431
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r7s x) H13R real_1 Hr7sI3).
L177431
We prove the intermediate claim HloI3: Rle one_third (apply_fun r7s x).
L177433
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_bounds).
L177433
We prove the intermediate claim HhiI3: Rle (apply_fun r7s x) 1.
L177435
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_bounds).
L177435
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) (minus_SNo one_third) ∈ R.
L177437
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) Hr7sRx (minus_SNo one_third) Hm13R).
L177437
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L177440
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r7s x) (minus_SNo one_third) H13R Hr7sRx Hm13R HloI3).
L177441
We prove the intermediate claim H13S: SNo one_third.
L177443
An exact proof term for the current goal is (real_SNo one_third H13R).
L177443
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L177445
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L177445
An exact proof term for the current goal is H0le_tmp.
L177446
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L177448
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L177448
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) (minus_SNo one_third)).
L177450
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r7s x) (minus_SNo one_third)) Hm23le0 H0le).
L177452
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r7s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L177455
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) 1 (minus_SNo one_third) Hr7sRx real_1 Hm13R HhiI3).
L177456
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L177458
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L177458
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L177459
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) (minus_SNo one_third)) two_thirds.
L177461
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L177464
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) (minus_SNo one_third)) Hm23R H23R Hr8xR Hlow Hup).
L177468
Assume HxnotC8: ¬ (x ∈ C8).
L177468
We prove the intermediate claim HxX: x ∈ X.
L177470
An exact proof term for the current goal is (HAsubX x HxA).
L177470
We prove the intermediate claim HnotI1: ¬ (apply_fun r7s x ∈ I1).
L177472
Assume Hr7sI1': apply_fun r7s x ∈ I1.
L177472
We prove the intermediate claim Hr7sI1I: apply_fun r7s x ∈ I1 ∩ I.
L177474
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r7s x) Hr7sI1' Hr7sIx).
L177474
We prove the intermediate claim HxB8': x ∈ B8.
L177476
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r7s x0 ∈ I1 ∩ I) x HxA Hr7sI1I).
L177476
Apply FalseE to the current goal.
L177477
An exact proof term for the current goal is (HxnotB8 HxB8').
L177478
We prove the intermediate claim HnotI3: ¬ (apply_fun r7s x ∈ I3).
L177480
Assume Hr7sI3': apply_fun r7s x ∈ I3.
L177480
We prove the intermediate claim Hr7sI3I: apply_fun r7s x ∈ I3 ∩ I.
L177482
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r7s x) Hr7sI3' Hr7sIx).
L177482
We prove the intermediate claim HxC8': x ∈ C8.
L177484
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r7s x0 ∈ I3 ∩ I) x HxA Hr7sI3I).
L177484
Apply FalseE to the current goal.
L177485
An exact proof term for the current goal is (HxnotC8 HxC8').
L177486
We prove the intermediate claim Hr7sRx: apply_fun r7s x ∈ R.
L177488
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r7s x) Hr7sIx).
L177488
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177490
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177490
We prove the intermediate claim H13R: one_third ∈ R.
L177492
An exact proof term for the current goal is one_third_in_R.
L177492
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177494
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177494
We prove the intermediate claim Hr7s_boundsI: Rle (minus_SNo 1) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) 1.
L177496
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r7s x) Hm1R real_1 Hr7sIx).
L177496
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r7s x) (minus_SNo 1)).
L177498
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r7s x) (andEL (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_boundsI)).
L177499
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r7s x)).
L177501
An exact proof term for the current goal is (RleE_nlt (apply_fun r7s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r7s x)) (Rle (apply_fun r7s x) 1) Hr7s_boundsI)).
L177502
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r7s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r7s x).
L177504
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r7s x) Hm1R Hm13R Hr7sRx HnotI1).
L177505
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r7s x).
L177507
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r7s x))) to the current goal.
L177508
Assume Hbad: Rlt (apply_fun r7s x) (minus_SNo 1).
L177508
Apply FalseE to the current goal.
L177509
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L177511
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r7s x).
L177511
An exact proof term for the current goal is Hok.
L177512
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r7s x) (minus_SNo one_third)).
L177514
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r7s x) Hm13lt_fx).
L177514
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r7s x) one_third ∨ Rlt 1 (apply_fun r7s x).
L177516
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r7s x) H13R real_1 Hr7sRx HnotI3).
L177517
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r7s x) one_third.
L177519
Apply (HnotI3_cases (Rlt (apply_fun r7s x) one_third)) to the current goal.
L177520
Assume Hok: Rlt (apply_fun r7s x) one_third.
L177520
An exact proof term for the current goal is Hok.
L177522
Assume Hbad: Rlt 1 (apply_fun r7s x).
L177522
Apply FalseE to the current goal.
L177523
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L177524
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r7s x)).
L177526
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r7s x) one_third Hfx_lt_13).
L177526
We prove the intermediate claim Hr7sI0: apply_fun r7s x ∈ I0.
L177528
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L177529
We prove the intermediate claim HxSep: apply_fun r7s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L177531
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r7s x) Hr7sRx (andI (¬ (Rlt (apply_fun r7s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r7s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L177535
rewrite the current goal using HI0_def (from left to right).
L177536
An exact proof term for the current goal is HxSep.
L177537
We prove the intermediate claim Hu8funI0: function_on u8 X I0.
L177539
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u8 Hu8contI0).
L177539
We prove the intermediate claim Hu8xI0: apply_fun u8 x ∈ I0.
L177541
An exact proof term for the current goal is (Hu8funI0 x HxX).
L177541
We prove the intermediate claim Hu8xR: apply_fun u8 x ∈ R.
L177543
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u8 x) Hu8xI0).
L177543
We prove the intermediate claim Hm_u8x_R: minus_SNo (apply_fun u8 x) ∈ R.
L177545
An exact proof term for the current goal is (real_minus_SNo (apply_fun u8 x) Hu8xR).
L177545
rewrite the current goal using (Hr8_apply x HxA) (from left to right).
L177546
We prove the intermediate claim Hr8xR: add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) ∈ R.
L177548
An exact proof term for the current goal is (real_add_SNo (apply_fun r7s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) (minus_SNo (apply_fun u8 x)) Hm_u8x_R).
L177550
We prove the intermediate claim H23R: two_thirds ∈ R.
L177552
An exact proof term for the current goal is two_thirds_in_R.
L177552
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177554
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177554
We prove the intermediate claim Hr7s_bounds0: Rle (minus_SNo one_third) (apply_fun r7s x) ∧ Rle (apply_fun r7s x) one_third.
L177556
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r7s x) Hm13R H13R Hr7sI0).
L177556
We prove the intermediate claim Hu8_bounds0: Rle (minus_SNo one_third) (apply_fun u8 x) ∧ Rle (apply_fun u8 x) one_third.
L177558
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u8 x) Hm13R H13R Hu8xI0).
L177558
We prove the intermediate claim Hm13_le_r7s: Rle (minus_SNo one_third) (apply_fun r7s x).
L177560
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r7s x)) (Rle (apply_fun r7s x) one_third) Hr7s_bounds0).
L177560
We prove the intermediate claim Hr7s_le_13: Rle (apply_fun r7s x) one_third.
L177562
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r7s x)) (Rle (apply_fun r7s x) one_third) Hr7s_bounds0).
L177562
We prove the intermediate claim Hm13_le_u8x: Rle (minus_SNo one_third) (apply_fun u8 x).
L177564
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u8 x)) (Rle (apply_fun u8 x) one_third) Hu8_bounds0).
L177564
We prove the intermediate claim Hu8x_le_13: Rle (apply_fun u8 x) one_third.
L177566
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u8 x)) (Rle (apply_fun u8 x) one_third) Hu8_bounds0).
L177566
We prove the intermediate claim Hm13_le_mu8: Rle (minus_SNo one_third) (minus_SNo (apply_fun u8 x)).
L177568
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u8 x) one_third Hu8x_le_13).
L177568
We prove the intermediate claim Hmu8_le_13: Rle (minus_SNo (apply_fun u8 x)) one_third.
L177570
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u8 x)) (minus_SNo (minus_SNo one_third)).
L177571
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u8 x) Hm13_le_u8x).
L177571
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L177572
An exact proof term for the current goal is Htmp.
L177573
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))).
L177576
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u8 x)) Hm13R Hm13R Hm_u8x_R Hm13_le_mu8).
L177577
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L177580
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) Hm_u8x_R Hm13_le_r7s).
L177584
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L177587
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) Hlow1 Hlow2).
L177590
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))).
L177592
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L177592
An exact proof term for the current goal is Hlow_tmp.
L177593
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) one_third).
L177596
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r7s x) (minus_SNo (apply_fun u8 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) Hm_u8x_R H13R Hmu8_le_13).
L177598
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r7s x) one_third) (add_SNo one_third one_third).
L177601
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r7s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r7s x) Hr7sI0) H13R H13R Hr7s_le_13).
L177603
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo one_third one_third).
L177606
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) (add_SNo (apply_fun r7s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L177609
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L177611
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) two_thirds.
L177613
rewrite the current goal using Hdef23 (from left to right) at position 1.
L177613
An exact proof term for the current goal is Hup_tmp.
L177614
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r7s x) (minus_SNo (apply_fun u8 x))) Hm23R H23R Hr8xR Hlow Hup).
L177617
Set r8s to be the term compose_fun A r8 (div_const_fun den).
L177618
We prove the intermediate claim Hr8s_cont: continuous_map A Ta I Ti r8s.
L177620
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L177621
An exact proof term for the current goal is R_standard_topology_is_topology.
L177621
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L177623
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L177623
We prove the intermediate claim Hr8s_contR: continuous_map A Ta R R_standard_topology r8s.
L177625
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r8 (div_const_fun den) Hr8_cont Hdivcont).
L177626
We prove the intermediate claim Hr8s_I: ∀x : set, x ∈ A → apply_fun r8s x ∈ I.
L177628
Let x be given.
L177628
Assume HxA: x ∈ A.
L177628
We prove the intermediate claim Hr8xI2: apply_fun r8 x ∈ I2.
L177630
An exact proof term for the current goal is (Hr8_range x HxA).
L177630
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L177632
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177632
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L177634
An exact proof term for the current goal is (real_minus_SNo den H23R).
L177634
We prove the intermediate claim Hr8xR: apply_fun r8 x ∈ R.
L177636
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r8 x) Hr8xI2).
L177636
We prove the intermediate claim Hr8xS: SNo (apply_fun r8 x).
L177638
An exact proof term for the current goal is (real_SNo (apply_fun r8 x) Hr8xR).
L177638
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r8 x) ∧ Rle (apply_fun r8 x) den.
L177640
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r8 x) HmdenR H23R Hr8xI2).
L177640
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r8 x).
L177642
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r8 x)) (Rle (apply_fun r8 x) den) Hbounds).
L177644
We prove the intermediate claim Hhi: Rle (apply_fun r8 x) den.
L177646
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r8 x)) (Rle (apply_fun r8 x) den) Hbounds).
L177648
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r8 x)).
L177650
An exact proof term for the current goal is (RleE_nlt (apply_fun r8 x) den Hhi).
L177650
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r8 x) (minus_SNo den)).
L177652
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r8 x) Hlo).
L177652
We prove the intermediate claim HyEq: apply_fun r8s x = div_SNo (apply_fun r8 x) den.
L177654
rewrite the current goal using (compose_fun_apply A r8 (div_const_fun den) x HxA) (from left to right).
L177654
rewrite the current goal using (div_const_fun_apply den (apply_fun r8 x) H23R Hr8xR) (from left to right).
Use reflexivity.
L177656
We prove the intermediate claim HyR: apply_fun r8s x ∈ R.
L177658
rewrite the current goal using HyEq (from left to right).
L177658
An exact proof term for the current goal is (real_div_SNo (apply_fun r8 x) Hr8xR den H23R).
L177659
We prove the intermediate claim HyS: SNo (apply_fun r8s x).
L177661
An exact proof term for the current goal is (real_SNo (apply_fun r8s x) HyR).
L177661
We prove the intermediate claim Hy_le_1: Rle (apply_fun r8s x) 1.
L177663
Apply (RleI (apply_fun r8s x) 1 HyR real_1) to the current goal.
L177663
We will prove ¬ (Rlt 1 (apply_fun r8s x)).
L177664
Assume H1lt: Rlt 1 (apply_fun r8s x).
L177665
We prove the intermediate claim H1lty: 1 < apply_fun r8s x.
L177667
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r8s x) H1lt).
L177667
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r8s x) den.
L177669
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r8s x) den SNo_1 HyS H23S H23pos H1lty).
L177669
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r8s x) den.
L177671
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L177671
An exact proof term for the current goal is HmulLt.
L177672
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r8s x) den = apply_fun r8 x.
L177674
rewrite the current goal using HyEq (from left to right) at position 1.
L177674
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r8 x) den Hr8xS H23S H23ne0).
L177675
We prove the intermediate claim Hden_lt_r8x: den < apply_fun r8 x.
L177677
rewrite the current goal using HmulEq (from right to left).
L177677
An exact proof term for the current goal is HmulLt'.
L177678
We prove the intermediate claim Hbad: Rlt den (apply_fun r8 x).
L177680
An exact proof term for the current goal is (RltI den (apply_fun r8 x) H23R Hr8xR Hden_lt_r8x).
L177680
An exact proof term for the current goal is (Hnlt_hi Hbad).
L177681
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r8s x).
L177683
Apply (RleI (minus_SNo 1) (apply_fun r8s x) Hm1R HyR) to the current goal.
L177683
We will prove ¬ (Rlt (apply_fun r8s x) (minus_SNo 1)).
L177684
Assume Hylt: Rlt (apply_fun r8s x) (minus_SNo 1).
L177685
We prove the intermediate claim Hylts: apply_fun r8s x < minus_SNo 1.
L177687
An exact proof term for the current goal is (RltE_lt (apply_fun r8s x) (minus_SNo 1) Hylt).
L177687
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r8s x) den < mul_SNo (minus_SNo 1) den.
L177689
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r8s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L177690
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r8s x) den = apply_fun r8 x.
L177692
rewrite the current goal using HyEq (from left to right) at position 1.
L177692
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r8 x) den Hr8xS H23S H23ne0).
L177693
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L177695
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L177695
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L177697
We prove the intermediate claim Hr8x_lt_mden: apply_fun r8 x < minus_SNo den.
L177699
rewrite the current goal using HmulEq (from right to left).
L177699
rewrite the current goal using HrhsEq (from right to left).
L177700
An exact proof term for the current goal is HmulLt.
L177701
We prove the intermediate claim Hbad: Rlt (apply_fun r8 x) (minus_SNo den).
L177703
An exact proof term for the current goal is (RltI (apply_fun r8 x) (minus_SNo den) Hr8xR HmdenR Hr8x_lt_mden).
L177703
An exact proof term for the current goal is (Hnlt_lo Hbad).
L177704
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r8s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L177706
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r8s I Hr8s_contR HIcR Hr8s_I).
L177707
We prove the intermediate claim Hex_u9: ∃u9 : set, continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x = one_third).
(*** fifteenth correction step scaffold: build u9 and g9 from residual r8s ***)
L177717
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r8s Hnorm HA Hr8s_cont).
L177717
Apply Hex_u9 to the current goal.
L177718
Let u9 be given.
L177719
Assume Hu9.
L177719
We prove the intermediate claim Hu9contI0: continuous_map X Tx I0 T0 u9.
L177721
We prove the intermediate claim Hu9AB: continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third).
L177725
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9 ∧ (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x = one_third) Hu9).
L177731
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9) (∀x : set, x ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x = minus_SNo one_third) Hu9AB).
L177736
We prove the intermediate claim Hu9contR: continuous_map X Tx R R_standard_topology u9.
L177738
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L177739
We prove the intermediate claim HI0subR: I0 ⊆ R.
L177741
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L177741
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u9 Hu9contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L177749
Set den9 to be the term mul_SNo den8 den.
L177750
We prove the intermediate claim Hden9R: den9 ∈ R.
L177752
An exact proof term for the current goal is (real_mul_SNo den8 Hden8R den H23R).
L177752
We prove the intermediate claim Hden9pos: 0 < den9.
L177754
We prove the intermediate claim Hden8S: SNo den8.
L177755
An exact proof term for the current goal is (real_SNo den8 Hden8R).
L177755
An exact proof term for the current goal is (mul_SNo_pos_pos den8 den Hden8S H23S Hden8pos HdenPos).
L177756
Set u9s to be the term compose_fun X u9 (mul_const_fun den9).
L177757
We prove the intermediate claim Hu9s_cont: continuous_map X Tx R R_standard_topology u9s.
L177759
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u9 den9 HTx Hu9contR Hden9R Hden9pos).
L177759
Set g9 to be the term compose_fun X (pair_map X g8 u9s) add_fun_R.
L177760
We prove the intermediate claim Hg9cont: continuous_map X Tx R R_standard_topology g9.
L177762
An exact proof term for the current goal is (add_two_continuous_R X Tx g8 u9s HTx Hg8cont Hu9s_cont).
L177762
We prove the intermediate claim Hu9contA: continuous_map A Ta R R_standard_topology u9.
(*** fifteenth correction step scaffold: residual r9 on A and scaling r9s ***)
L177766
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u9 A HTx HAsubX Hu9contR).
L177766
Set u9neg to be the term compose_fun A u9 neg_fun.
L177767
We prove the intermediate claim Hu9neg_cont: continuous_map A Ta R R_standard_topology u9neg.
L177769
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u9 neg_fun Hu9contA Hnegcont).
L177770
We prove the intermediate claim Hr8s_contR: continuous_map A Ta R R_standard_topology r8s.
L177772
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r8s Hr8s_cont HIcR R_standard_topology_is_topology_local HTiEq).
L177773
Set r9 to be the term compose_fun A (pair_map A r8s u9neg) add_fun_R.
L177774
We prove the intermediate claim Hr9_cont: continuous_map A Ta R R_standard_topology r9.
L177776
An exact proof term for the current goal is (add_two_continuous_R A Ta r8s u9neg HTa Hr8s_contR Hu9neg_cont).
L177776
We prove the intermediate claim Hr9_apply: ∀x : set, x ∈ A → apply_fun r9 x = add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x)).
L177779
Let x be given.
L177779
Assume HxA: x ∈ A.
L177779
We prove the intermediate claim Hpimg: apply_fun (pair_map A r8s u9neg) x ∈ setprod R R.
L177781
rewrite the current goal using (pair_map_apply A R R r8s u9neg x HxA) (from left to right).
L177781
We prove the intermediate claim Hr8sxI: apply_fun r8s x ∈ I.
L177783
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r8s Hr8s_cont x HxA).
L177783
We prove the intermediate claim Hr8sxR: apply_fun r8s x ∈ R.
L177785
An exact proof term for the current goal is (HIcR (apply_fun r8s x) Hr8sxI).
L177785
We prove the intermediate claim Hu9negRx: apply_fun u9neg x ∈ R.
L177787
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u9neg Hu9neg_cont x HxA).
L177787
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r8s x) (apply_fun u9neg x) Hr8sxR Hu9negRx).
L177788
rewrite the current goal using (compose_fun_apply A (pair_map A r8s u9neg) add_fun_R x HxA) (from left to right).
L177789
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r8s u9neg) x) Hpimg) (from left to right) at position 1.
L177790
rewrite the current goal using (pair_map_apply A R R r8s u9neg x HxA) (from left to right).
L177791
rewrite the current goal using (tuple_2_0_eq (apply_fun r8s x) (apply_fun u9neg x)) (from left to right).
L177792
rewrite the current goal using (tuple_2_1_eq (apply_fun r8s x) (apply_fun u9neg x)) (from left to right).
L177793
rewrite the current goal using (compose_fun_apply A u9 neg_fun x HxA) (from left to right) at position 1.
L177794
We prove the intermediate claim Hu9Rx: apply_fun u9 x ∈ R.
L177796
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u9 Hu9contA x HxA).
L177796
rewrite the current goal using (neg_fun_apply (apply_fun u9 x) Hu9Rx) (from left to right) at position 1.
Use reflexivity.
L177798
We prove the intermediate claim Hr9_range: ∀x : set, x ∈ A → apply_fun r9 x ∈ I2.
L177800
We prove the intermediate claim Hu9AB: continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third).
L177804
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third) Hu9).
L177810
We prove the intermediate claim Hu9_on_B9: ∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third.
L177814
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u9) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third) Hu9AB).
L177818
We prove the intermediate claim Hu9_on_C9: ∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third.
L177822
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u9 ∧ (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u9 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r8s ((closed_interval one_third 1) ∩ I) → apply_fun u9 x0 = one_third) Hu9).
L177828
Let x be given.
L177829
Assume HxA: x ∈ A.
L177829
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L177830
Set I3 to be the term closed_interval one_third 1.
L177831
Set B9 to be the term preimage_of A r8s (I1 ∩ I).
L177832
Set C9 to be the term preimage_of A r8s (I3 ∩ I).
L177833
We prove the intermediate claim Hr8sIx: apply_fun r8s x ∈ I.
L177835
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r8s Hr8s_cont x HxA).
L177835
We prove the intermediate claim HB9_cases: x ∈ B9 ∨ ¬ (x ∈ B9).
L177837
An exact proof term for the current goal is (xm (x ∈ B9)).
L177837
Apply (HB9_cases (apply_fun r9 x ∈ I2)) to the current goal.
L177839
Assume HxB9: x ∈ B9.
L177839
We prove the intermediate claim Hu9eq: apply_fun u9 x = minus_SNo one_third.
L177841
An exact proof term for the current goal is (Hu9_on_B9 x HxB9).
L177841
We prove the intermediate claim Hr9eq: apply_fun r9 x = add_SNo (apply_fun r8s x) one_third.
L177843
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L177843
rewrite the current goal using Hu9eq (from left to right) at position 1.
L177844
We prove the intermediate claim H13R: one_third ∈ R.
L177846
An exact proof term for the current goal is one_third_in_R.
L177846
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L177848
rewrite the current goal using Hr9eq (from left to right).
L177849
We prove the intermediate claim Hr8sI1I: apply_fun r8s x ∈ I1 ∩ I.
L177851
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r8s x0 ∈ I1 ∩ I) x HxB9).
L177851
We prove the intermediate claim Hr8sI1: apply_fun r8s x ∈ I1.
L177853
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r8s x) Hr8sI1I).
L177853
We prove the intermediate claim H13R: one_third ∈ R.
L177855
An exact proof term for the current goal is one_third_in_R.
L177855
We prove the intermediate claim H23R: two_thirds ∈ R.
L177857
An exact proof term for the current goal is two_thirds_in_R.
L177857
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177859
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177859
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177861
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177861
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177863
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177863
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L177865
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hr8sI1).
L177865
We prove the intermediate claim Hr8s_bounds: Rle (minus_SNo 1) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) (minus_SNo one_third).
L177867
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hm1R Hm13R Hr8sI1).
L177868
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r8s x).
L177870
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) (minus_SNo one_third)) Hr8s_bounds).
L177871
We prove the intermediate claim HhiI1: Rle (apply_fun r8s x) (minus_SNo one_third).
L177873
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) (minus_SNo one_third)) Hr8s_bounds).
L177874
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) one_third ∈ R.
L177876
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) Hr8sRx one_third H13R).
L177876
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r8s x) one_third).
L177878
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r8s x) one_third Hm1R Hr8sRx H13R Hm1le).
L177878
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) one_third).
L177880
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L177880
An exact proof term for the current goal is Hlow_tmp.
L177881
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r8s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L177883
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) (minus_SNo one_third) one_third Hr8sRx Hm13R H13R HhiI1).
L177883
We prove the intermediate claim H13S: SNo one_third.
L177885
An exact proof term for the current goal is (real_SNo one_third H13R).
L177885
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r8s x) one_third) 0.
L177887
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L177887
An exact proof term for the current goal is Hup0_tmp.
L177888
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L177890
An exact proof term for the current goal is Rle_0_two_thirds.
L177890
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) one_third) two_thirds.
L177892
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) one_third) 0 two_thirds Hup0 H0le23).
L177892
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) one_third) Hm23R H23R Hr9xR Hlow Hup).
L177896
Assume HxnotB9: ¬ (x ∈ B9).
L177896
We prove the intermediate claim HC9_cases: x ∈ C9 ∨ ¬ (x ∈ C9).
L177898
An exact proof term for the current goal is (xm (x ∈ C9)).
L177898
Apply (HC9_cases (apply_fun r9 x ∈ I2)) to the current goal.
L177900
Assume HxC9: x ∈ C9.
L177900
We prove the intermediate claim Hu9eq: apply_fun u9 x = one_third.
L177902
An exact proof term for the current goal is (Hu9_on_C9 x HxC9).
L177902
We prove the intermediate claim Hr9eq: apply_fun r9 x = add_SNo (apply_fun r8s x) (minus_SNo one_third).
L177904
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L177904
rewrite the current goal using Hu9eq (from left to right) at position 1.
Use reflexivity.
L177906
rewrite the current goal using Hr9eq (from left to right).
L177907
We prove the intermediate claim Hr8sI3I: apply_fun r8s x ∈ I3 ∩ I.
L177909
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r8s x0 ∈ I3 ∩ I) x HxC9).
L177909
We prove the intermediate claim Hr8sI3: apply_fun r8s x ∈ I3.
L177911
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r8s x) Hr8sI3I).
L177911
We prove the intermediate claim H13R: one_third ∈ R.
L177913
An exact proof term for the current goal is one_third_in_R.
L177913
We prove the intermediate claim H23R: two_thirds ∈ R.
L177915
An exact proof term for the current goal is two_thirds_in_R.
L177915
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L177917
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L177917
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177919
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177919
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L177921
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r8s x) Hr8sI3).
L177921
We prove the intermediate claim Hr8s_bounds: Rle one_third (apply_fun r8s x) ∧ Rle (apply_fun r8s x) 1.
L177923
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r8s x) H13R real_1 Hr8sI3).
L177923
We prove the intermediate claim HloI3: Rle one_third (apply_fun r8s x).
L177925
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_bounds).
L177925
We prove the intermediate claim HhiI3: Rle (apply_fun r8s x) 1.
L177927
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_bounds).
L177927
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) (minus_SNo one_third) ∈ R.
L177929
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) Hr8sRx (minus_SNo one_third) Hm13R).
L177929
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L177932
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r8s x) (minus_SNo one_third) H13R Hr8sRx Hm13R HloI3).
L177933
We prove the intermediate claim H13S: SNo one_third.
L177935
An exact proof term for the current goal is (real_SNo one_third H13R).
L177935
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L177937
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L177937
An exact proof term for the current goal is H0le_tmp.
L177938
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L177940
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L177940
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) (minus_SNo one_third)).
L177942
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r8s x) (minus_SNo one_third)) Hm23le0 H0le).
L177944
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r8s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L177947
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) 1 (minus_SNo one_third) Hr8sRx real_1 Hm13R HhiI3).
L177948
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L177950
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L177950
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L177951
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) (minus_SNo one_third)) two_thirds.
L177953
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L177956
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) (minus_SNo one_third)) Hm23R H23R Hr9xR Hlow Hup).
L177960
Assume HxnotC9: ¬ (x ∈ C9).
L177960
We prove the intermediate claim HxX: x ∈ X.
L177962
An exact proof term for the current goal is (HAsubX x HxA).
L177962
We prove the intermediate claim HnotI1: ¬ (apply_fun r8s x ∈ I1).
L177964
Assume Hr8sI1': apply_fun r8s x ∈ I1.
L177964
We prove the intermediate claim Hr8sI1I: apply_fun r8s x ∈ I1 ∩ I.
L177966
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r8s x) Hr8sI1' Hr8sIx).
L177966
We prove the intermediate claim HxB9': x ∈ B9.
L177968
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r8s x0 ∈ I1 ∩ I) x HxA Hr8sI1I).
L177968
Apply FalseE to the current goal.
L177969
An exact proof term for the current goal is (HxnotB9 HxB9').
L177970
We prove the intermediate claim HnotI3: ¬ (apply_fun r8s x ∈ I3).
L177972
Assume Hr8sI3': apply_fun r8s x ∈ I3.
L177972
We prove the intermediate claim Hr8sI3I: apply_fun r8s x ∈ I3 ∩ I.
L177974
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r8s x) Hr8sI3' Hr8sIx).
L177974
We prove the intermediate claim HxC9': x ∈ C9.
L177976
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r8s x0 ∈ I3 ∩ I) x HxA Hr8sI3I).
L177976
Apply FalseE to the current goal.
L177977
An exact proof term for the current goal is (HxnotC9 HxC9').
L177978
We prove the intermediate claim Hr8sRx: apply_fun r8s x ∈ R.
L177980
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r8s x) Hr8sIx).
L177980
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L177982
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L177982
We prove the intermediate claim H13R: one_third ∈ R.
L177984
An exact proof term for the current goal is one_third_in_R.
L177984
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L177986
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L177986
We prove the intermediate claim Hr8s_boundsI: Rle (minus_SNo 1) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) 1.
L177988
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r8s x) Hm1R real_1 Hr8sIx).
L177988
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r8s x) (minus_SNo 1)).
L177990
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r8s x) (andEL (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_boundsI)).
L177991
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r8s x)).
L177993
An exact proof term for the current goal is (RleE_nlt (apply_fun r8s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r8s x)) (Rle (apply_fun r8s x) 1) Hr8s_boundsI)).
L177994
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r8s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r8s x).
L177996
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r8s x) Hm1R Hm13R Hr8sRx HnotI1).
L177997
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r8s x).
L177999
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r8s x))) to the current goal.
L178000
Assume Hbad: Rlt (apply_fun r8s x) (minus_SNo 1).
L178000
Apply FalseE to the current goal.
L178001
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L178003
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r8s x).
L178003
An exact proof term for the current goal is Hok.
L178004
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r8s x) (minus_SNo one_third)).
L178006
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r8s x) Hm13lt_fx).
L178006
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r8s x) one_third ∨ Rlt 1 (apply_fun r8s x).
L178008
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r8s x) H13R real_1 Hr8sRx HnotI3).
L178009
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r8s x) one_third.
L178011
Apply (HnotI3_cases (Rlt (apply_fun r8s x) one_third)) to the current goal.
L178012
Assume Hok: Rlt (apply_fun r8s x) one_third.
L178012
An exact proof term for the current goal is Hok.
L178014
Assume Hbad: Rlt 1 (apply_fun r8s x).
L178014
Apply FalseE to the current goal.
L178015
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L178016
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r8s x)).
L178018
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r8s x) one_third Hfx_lt_13).
L178018
We prove the intermediate claim Hr8sI0: apply_fun r8s x ∈ I0.
L178020
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L178021
We prove the intermediate claim HxSep: apply_fun r8s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L178023
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r8s x) Hr8sRx (andI (¬ (Rlt (apply_fun r8s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r8s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L178027
rewrite the current goal using HI0_def (from left to right).
L178028
An exact proof term for the current goal is HxSep.
L178029
We prove the intermediate claim Hu9funI0: function_on u9 X I0.
L178031
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u9 Hu9contI0).
L178031
We prove the intermediate claim Hu9xI0: apply_fun u9 x ∈ I0.
L178033
An exact proof term for the current goal is (Hu9funI0 x HxX).
L178033
We prove the intermediate claim Hu9xR: apply_fun u9 x ∈ R.
L178035
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u9 x) Hu9xI0).
L178035
We prove the intermediate claim Hm_u9x_R: minus_SNo (apply_fun u9 x) ∈ R.
L178037
An exact proof term for the current goal is (real_minus_SNo (apply_fun u9 x) Hu9xR).
L178037
rewrite the current goal using (Hr9_apply x HxA) (from left to right).
L178038
We prove the intermediate claim Hr9xR: add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) ∈ R.
L178040
An exact proof term for the current goal is (real_add_SNo (apply_fun r8s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) (minus_SNo (apply_fun u9 x)) Hm_u9x_R).
L178042
We prove the intermediate claim H23R: two_thirds ∈ R.
L178044
An exact proof term for the current goal is two_thirds_in_R.
L178044
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178046
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178046
We prove the intermediate claim Hr8s_bounds0: Rle (minus_SNo one_third) (apply_fun r8s x) ∧ Rle (apply_fun r8s x) one_third.
L178048
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r8s x) Hm13R H13R Hr8sI0).
L178048
We prove the intermediate claim Hu9_bounds0: Rle (minus_SNo one_third) (apply_fun u9 x) ∧ Rle (apply_fun u9 x) one_third.
L178050
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u9 x) Hm13R H13R Hu9xI0).
L178050
We prove the intermediate claim Hm13_le_r8s: Rle (minus_SNo one_third) (apply_fun r8s x).
L178052
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r8s x)) (Rle (apply_fun r8s x) one_third) Hr8s_bounds0).
L178052
We prove the intermediate claim Hr8s_le_13: Rle (apply_fun r8s x) one_third.
L178054
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r8s x)) (Rle (apply_fun r8s x) one_third) Hr8s_bounds0).
L178054
We prove the intermediate claim Hm13_le_u9x: Rle (minus_SNo one_third) (apply_fun u9 x).
L178056
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u9 x)) (Rle (apply_fun u9 x) one_third) Hu9_bounds0).
L178056
We prove the intermediate claim Hu9x_le_13: Rle (apply_fun u9 x) one_third.
L178058
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u9 x)) (Rle (apply_fun u9 x) one_third) Hu9_bounds0).
L178058
We prove the intermediate claim Hm13_le_mu9: Rle (minus_SNo one_third) (minus_SNo (apply_fun u9 x)).
L178060
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u9 x) one_third Hu9x_le_13).
L178060
We prove the intermediate claim Hmu9_le_13: Rle (minus_SNo (apply_fun u9 x)) one_third.
L178062
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u9 x)) (minus_SNo (minus_SNo one_third)).
L178063
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u9 x) Hm13_le_u9x).
L178063
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L178064
An exact proof term for the current goal is Htmp.
L178065
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))).
L178068
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u9 x)) Hm13R Hm13R Hm_u9x_R Hm13_le_mu9).
L178069
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L178072
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) Hm_u9x_R Hm13_le_r8s).
L178076
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L178079
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) Hlow1 Hlow2).
L178082
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))).
L178084
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L178084
An exact proof term for the current goal is Hlow_tmp.
L178085
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) one_third).
L178088
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r8s x) (minus_SNo (apply_fun u9 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) Hm_u9x_R H13R Hmu9_le_13).
L178090
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r8s x) one_third) (add_SNo one_third one_third).
L178093
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r8s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r8s x) Hr8sI0) H13R H13R Hr8s_le_13).
L178095
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo one_third one_third).
L178098
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) (add_SNo (apply_fun r8s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L178101
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L178103
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) two_thirds.
L178105
rewrite the current goal using Hdef23 (from left to right) at position 1.
L178105
An exact proof term for the current goal is Hup_tmp.
L178106
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r8s x) (minus_SNo (apply_fun u9 x))) Hm23R H23R Hr9xR Hlow Hup).
L178109
Set r9s to be the term compose_fun A r9 (div_const_fun den).
L178110
We prove the intermediate claim Hr9s_cont: continuous_map A Ta I Ti r9s.
L178112
We prove the intermediate claim HRtop: topology_on R R_standard_topology.
L178113
An exact proof term for the current goal is R_standard_topology_is_topology.
L178113
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L178115
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L178115
We prove the intermediate claim Hr9s_contR: continuous_map A Ta R R_standard_topology r9s.
L178117
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r9 (div_const_fun den) Hr9_cont Hdivcont).
L178118
We prove the intermediate claim Hr9s_I: ∀x : set, x ∈ A → apply_fun r9s x ∈ I.
L178120
Let x be given.
L178120
Assume HxA: x ∈ A.
L178120
We prove the intermediate claim Hr9xI2: apply_fun r9 x ∈ I2.
L178122
An exact proof term for the current goal is (Hr9_range x HxA).
L178122
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L178124
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178124
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L178126
An exact proof term for the current goal is (real_minus_SNo den H23R).
L178126
We prove the intermediate claim Hr9xR: apply_fun r9 x ∈ R.
L178128
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r9 x) Hr9xI2).
L178128
We prove the intermediate claim Hr9xS: SNo (apply_fun r9 x).
L178130
An exact proof term for the current goal is (real_SNo (apply_fun r9 x) Hr9xR).
L178130
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r9 x) ∧ Rle (apply_fun r9 x) den.
L178132
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r9 x) HmdenR H23R Hr9xI2).
L178132
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r9 x).
L178134
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r9 x)) (Rle (apply_fun r9 x) den) Hbounds).
L178136
We prove the intermediate claim Hhi: Rle (apply_fun r9 x) den.
L178138
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r9 x)) (Rle (apply_fun r9 x) den) Hbounds).
L178140
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r9 x)).
L178142
An exact proof term for the current goal is (RleE_nlt (apply_fun r9 x) den Hhi).
L178142
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r9 x) (minus_SNo den)).
L178144
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r9 x) Hlo).
L178144
We prove the intermediate claim HyEq: apply_fun r9s x = div_SNo (apply_fun r9 x) den.
L178146
rewrite the current goal using (compose_fun_apply A r9 (div_const_fun den) x HxA) (from left to right).
L178146
rewrite the current goal using (div_const_fun_apply den (apply_fun r9 x) H23R Hr9xR) (from left to right).
Use reflexivity.
L178148
We prove the intermediate claim HyR: apply_fun r9s x ∈ R.
L178150
rewrite the current goal using HyEq (from left to right).
L178150
An exact proof term for the current goal is (real_div_SNo (apply_fun r9 x) Hr9xR den H23R).
L178151
We prove the intermediate claim HyS: SNo (apply_fun r9s x).
L178153
An exact proof term for the current goal is (real_SNo (apply_fun r9s x) HyR).
L178153
We prove the intermediate claim Hy_le_1: Rle (apply_fun r9s x) 1.
L178155
Apply (RleI (apply_fun r9s x) 1 HyR real_1) to the current goal.
L178155
We will prove ¬ (Rlt 1 (apply_fun r9s x)).
L178156
Assume H1lt: Rlt 1 (apply_fun r9s x).
L178157
We prove the intermediate claim H1lty: 1 < apply_fun r9s x.
L178159
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r9s x) H1lt).
L178159
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r9s x) den.
L178161
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r9s x) den SNo_1 HyS H23S H23pos H1lty).
L178161
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r9s x) den.
L178163
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L178163
An exact proof term for the current goal is HmulLt.
L178164
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r9s x) den = apply_fun r9 x.
L178166
rewrite the current goal using HyEq (from left to right) at position 1.
L178166
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r9 x) den Hr9xS H23S H23ne0).
L178167
We prove the intermediate claim Hden_lt_r9x: den < apply_fun r9 x.
L178169
rewrite the current goal using HmulEq (from right to left).
L178169
An exact proof term for the current goal is HmulLt'.
L178170
We prove the intermediate claim Hbad: Rlt den (apply_fun r9 x).
L178172
An exact proof term for the current goal is (RltI den (apply_fun r9 x) H23R Hr9xR Hden_lt_r9x).
L178172
An exact proof term for the current goal is (Hnlt_hi Hbad).
L178173
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r9s x).
L178175
Apply (RleI (minus_SNo 1) (apply_fun r9s x) Hm1R HyR) to the current goal.
L178175
We will prove ¬ (Rlt (apply_fun r9s x) (minus_SNo 1)).
L178176
Assume Hylt: Rlt (apply_fun r9s x) (minus_SNo 1).
L178177
We prove the intermediate claim Hylts: apply_fun r9s x < minus_SNo 1.
L178179
An exact proof term for the current goal is (RltE_lt (apply_fun r9s x) (minus_SNo 1) Hylt).
L178179
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r9s x) den < mul_SNo (minus_SNo 1) den.
L178181
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r9s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L178182
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r9s x) den = apply_fun r9 x.
L178184
rewrite the current goal using HyEq (from left to right) at position 1.
L178184
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r9 x) den Hr9xS H23S H23ne0).
L178185
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L178187
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L178187
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L178189
We prove the intermediate claim Hr9x_lt_mden: apply_fun r9 x < minus_SNo den.
L178191
rewrite the current goal using HmulEq (from right to left).
L178191
rewrite the current goal using HrhsEq (from right to left).
L178192
An exact proof term for the current goal is HmulLt.
L178193
We prove the intermediate claim Hbad: Rlt (apply_fun r9 x) (minus_SNo den).
L178195
An exact proof term for the current goal is (RltI (apply_fun r9 x) (minus_SNo den) Hr9xR HmdenR Hr9x_lt_mden).
L178195
An exact proof term for the current goal is (Hnlt_lo Hbad).
L178196
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r9s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L178198
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r9s I Hr9s_contR HIcR Hr9s_I).
L178199
We prove the intermediate claim Hex_u10: ∃u10 : set, continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x = one_third).
(*** further iteration scaffold: build u10 and g10 from residual r9s ***)
L178209
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r9s Hnorm HA Hr9s_cont).
L178209
Apply Hex_u10 to the current goal.
L178210
Let u10 be given.
L178211
Assume Hu10.
L178211
We prove the intermediate claim Hu10contI0: continuous_map X Tx I0 T0 u10.
L178213
We prove the intermediate claim Hu10AB: continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third).
L178217
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10 ∧ (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x = one_third) Hu10).
L178223
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10) (∀x : set, x ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x = minus_SNo one_third) Hu10AB).
L178228
We prove the intermediate claim Hu10contR: continuous_map X Tx R R_standard_topology u10.
L178230
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L178231
We prove the intermediate claim HI0subR: I0 ⊆ R.
L178233
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L178233
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u10 Hu10contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L178241
Set den10 to be the term mul_SNo den9 den.
L178242
We prove the intermediate claim Hden10R: den10 ∈ R.
L178244
An exact proof term for the current goal is (real_mul_SNo den9 Hden9R den H23R).
L178244
We prove the intermediate claim Hden10pos: 0 < den10.
L178246
We prove the intermediate claim Hden9S: SNo den9.
L178247
An exact proof term for the current goal is (real_SNo den9 Hden9R).
L178247
An exact proof term for the current goal is (mul_SNo_pos_pos den9 den Hden9S H23S Hden9pos HdenPos).
L178248
Set u10s to be the term compose_fun X u10 (mul_const_fun den10).
L178249
We prove the intermediate claim Hu10s_cont: continuous_map X Tx R R_standard_topology u10s.
L178251
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u10 den10 HTx Hu10contR Hden10R Hden10pos).
L178251
Set g10 to be the term compose_fun X (pair_map X g9 u10s) add_fun_R.
L178252
We prove the intermediate claim Hg10cont: continuous_map X Tx R R_standard_topology g10.
L178254
An exact proof term for the current goal is (add_two_continuous_R X Tx g9 u10s HTx Hg9cont Hu10s_cont).
L178254
We prove the intermediate claim Hu10contA: continuous_map A Ta R R_standard_topology u10.
(*** further iteration scaffold: residual r10 on A and scaling r10s ***)
L178258
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u10 A HTx HAsubX Hu10contR).
L178258
Set u10neg to be the term compose_fun A u10 neg_fun.
L178259
We prove the intermediate claim Hu10neg_cont: continuous_map A Ta R R_standard_topology u10neg.
L178261
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u10 neg_fun Hu10contA Hnegcont).
L178262
We prove the intermediate claim Hr9s_contR: continuous_map A Ta R R_standard_topology r9s.
L178264
We prove the intermediate claim HTiEq': Ti = subspace_topology R R_standard_topology I.
L178265
An exact proof term for the current goal is HTiEq.
L178265
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r9s Hr9s_cont HIcR R_standard_topology_is_topology_local HTiEq').
L178267
Set r10 to be the term compose_fun A (pair_map A r9s u10neg) add_fun_R.
L178268
We prove the intermediate claim Hr10_cont: continuous_map A Ta R R_standard_topology r10.
L178270
An exact proof term for the current goal is (add_two_continuous_R A Ta r9s u10neg HTa Hr9s_contR Hu10neg_cont).
L178270
We prove the intermediate claim Hr10_apply: ∀x : set, x ∈ A → apply_fun r10 x = add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x)).
L178273
Let x be given.
L178273
Assume HxA: x ∈ A.
L178273
We prove the intermediate claim Hpimg: apply_fun (pair_map A r9s u10neg) x ∈ setprod R R.
L178275
rewrite the current goal using (pair_map_apply A R R r9s u10neg x HxA) (from left to right).
L178275
We prove the intermediate claim Hr9sxI: apply_fun r9s x ∈ I.
L178277
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r9s Hr9s_cont x HxA).
L178277
We prove the intermediate claim Hr9sxR: apply_fun r9s x ∈ R.
L178279
An exact proof term for the current goal is (HIcR (apply_fun r9s x) Hr9sxI).
L178279
We prove the intermediate claim Hu10negRx: apply_fun u10neg x ∈ R.
L178281
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u10neg Hu10neg_cont x HxA).
L178281
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r9s x) (apply_fun u10neg x) Hr9sxR Hu10negRx).
L178282
rewrite the current goal using (compose_fun_apply A (pair_map A r9s u10neg) add_fun_R x HxA) (from left to right).
L178283
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r9s u10neg) x) Hpimg) (from left to right) at position 1.
L178284
rewrite the current goal using (pair_map_apply A R R r9s u10neg x HxA) (from left to right).
L178285
rewrite the current goal using (tuple_2_0_eq (apply_fun r9s x) (apply_fun u10neg x)) (from left to right).
L178286
rewrite the current goal using (tuple_2_1_eq (apply_fun r9s x) (apply_fun u10neg x)) (from left to right).
L178287
rewrite the current goal using (compose_fun_apply A u10 neg_fun x HxA) (from left to right).
L178288
We prove the intermediate claim Hu10Rx: apply_fun u10 x ∈ R.
L178290
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u10 Hu10contA x HxA).
L178290
rewrite the current goal using (neg_fun_apply (apply_fun u10 x) Hu10Rx) (from left to right).
Use reflexivity.
L178292
We prove the intermediate claim Hr10_range: ∀x : set, x ∈ A → apply_fun r10 x ∈ I2.
L178294
We prove the intermediate claim Hu10AB: continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third).
L178298
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third) Hu10).
L178304
We prove the intermediate claim Hu10_on_B10: ∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third.
L178308
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u10) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third) Hu10AB).
L178312
We prove the intermediate claim Hu10_on_C10: ∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third.
L178316
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u10 ∧ (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u10 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r9s ((closed_interval one_third 1) ∩ I) → apply_fun u10 x0 = one_third) Hu10).
L178322
Let x be given.
L178323
Assume HxA: x ∈ A.
L178323
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L178324
Set I3 to be the term closed_interval one_third 1.
L178325
Set B10 to be the term preimage_of A r9s (I1 ∩ I).
L178326
Set C10 to be the term preimage_of A r9s (I3 ∩ I).
L178327
We prove the intermediate claim Hr9sIx: apply_fun r9s x ∈ I.
L178329
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r9s Hr9s_cont x HxA).
L178329
We prove the intermediate claim HB10_cases: x ∈ B10 ∨ ¬ (x ∈ B10).
L178331
An exact proof term for the current goal is (xm (x ∈ B10)).
L178331
Apply (HB10_cases (apply_fun r10 x ∈ I2)) to the current goal.
L178333
Assume HxB10: x ∈ B10.
L178333
We prove the intermediate claim Hu10eq: apply_fun u10 x = minus_SNo one_third.
L178335
An exact proof term for the current goal is (Hu10_on_B10 x HxB10).
L178335
We prove the intermediate claim Hr10eq: apply_fun r10 x = add_SNo (apply_fun r9s x) one_third.
L178337
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L178337
rewrite the current goal using Hu10eq (from left to right) at position 1.
L178338
We prove the intermediate claim H13R: one_third ∈ R.
L178340
An exact proof term for the current goal is one_third_in_R.
L178340
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L178342
rewrite the current goal using Hr10eq (from left to right).
L178343
We prove the intermediate claim Hr9sI1I: apply_fun r9s x ∈ I1 ∩ I.
L178345
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r9s x0 ∈ I1 ∩ I) x HxB10).
L178345
We prove the intermediate claim Hr9sI1: apply_fun r9s x ∈ I1.
L178347
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r9s x) Hr9sI1I).
L178347
We prove the intermediate claim H13R: one_third ∈ R.
L178349
An exact proof term for the current goal is one_third_in_R.
L178349
We prove the intermediate claim H23R: two_thirds ∈ R.
L178351
An exact proof term for the current goal is two_thirds_in_R.
L178351
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178353
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178353
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178355
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178355
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178357
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178357
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L178359
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hr9sI1).
L178359
We prove the intermediate claim Hr9s_bounds: Rle (minus_SNo 1) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) (minus_SNo one_third).
L178361
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hm1R Hm13R Hr9sI1).
L178362
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r9s x).
L178364
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) (minus_SNo one_third)) Hr9s_bounds).
L178365
We prove the intermediate claim HhiI1: Rle (apply_fun r9s x) (minus_SNo one_third).
L178367
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) (minus_SNo one_third)) Hr9s_bounds).
L178368
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) one_third ∈ R.
L178370
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) Hr9sRx one_third H13R).
L178370
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r9s x) one_third).
L178372
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r9s x) one_third Hm1R Hr9sRx H13R Hm1le).
L178372
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) one_third).
L178374
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L178374
An exact proof term for the current goal is Hlow_tmp.
L178375
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r9s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L178377
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) (minus_SNo one_third) one_third Hr9sRx Hm13R H13R HhiI1).
L178377
We prove the intermediate claim H13S: SNo one_third.
L178379
An exact proof term for the current goal is (real_SNo one_third H13R).
L178379
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r9s x) one_third) 0.
L178381
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L178381
An exact proof term for the current goal is Hup0_tmp.
L178382
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L178384
An exact proof term for the current goal is Rle_0_two_thirds.
L178384
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) one_third) two_thirds.
L178386
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) one_third) 0 two_thirds Hup0 H0le23).
L178386
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) one_third) Hm23R H23R Hr10xR Hlow Hup).
L178390
Assume HxnotB10: ¬ (x ∈ B10).
L178390
We prove the intermediate claim HC10_cases: x ∈ C10 ∨ ¬ (x ∈ C10).
L178392
An exact proof term for the current goal is (xm (x ∈ C10)).
L178392
Apply (HC10_cases (apply_fun r10 x ∈ I2)) to the current goal.
L178394
Assume HxC10: x ∈ C10.
L178394
We prove the intermediate claim Hu10eq: apply_fun u10 x = one_third.
L178396
An exact proof term for the current goal is (Hu10_on_C10 x HxC10).
L178396
We prove the intermediate claim Hr10eq: apply_fun r10 x = add_SNo (apply_fun r9s x) (minus_SNo one_third).
L178398
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L178398
rewrite the current goal using Hu10eq (from left to right) at position 1.
Use reflexivity.
L178400
rewrite the current goal using Hr10eq (from left to right).
L178401
We prove the intermediate claim Hr9sI3I: apply_fun r9s x ∈ I3 ∩ I.
L178403
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r9s x0 ∈ I3 ∩ I) x HxC10).
L178403
We prove the intermediate claim Hr9sI3: apply_fun r9s x ∈ I3.
L178405
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r9s x) Hr9sI3I).
L178405
We prove the intermediate claim H13R: one_third ∈ R.
L178407
An exact proof term for the current goal is one_third_in_R.
L178407
We prove the intermediate claim H23R: two_thirds ∈ R.
L178409
An exact proof term for the current goal is two_thirds_in_R.
L178409
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178411
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178411
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178413
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178413
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L178415
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r9s x) Hr9sI3).
L178415
We prove the intermediate claim Hr9s_bounds: Rle one_third (apply_fun r9s x) ∧ Rle (apply_fun r9s x) 1.
L178417
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r9s x) H13R real_1 Hr9sI3).
L178417
We prove the intermediate claim HloI3: Rle one_third (apply_fun r9s x).
L178419
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_bounds).
L178419
We prove the intermediate claim HhiI3: Rle (apply_fun r9s x) 1.
L178421
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_bounds).
L178421
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) (minus_SNo one_third) ∈ R.
L178423
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) Hr9sRx (minus_SNo one_third) Hm13R).
L178423
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L178426
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r9s x) (minus_SNo one_third) H13R Hr9sRx Hm13R HloI3).
L178427
We prove the intermediate claim H13S: SNo one_third.
L178429
An exact proof term for the current goal is (real_SNo one_third H13R).
L178429
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L178431
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L178431
An exact proof term for the current goal is H0le_tmp.
L178432
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L178434
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L178434
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) (minus_SNo one_third)).
L178436
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r9s x) (minus_SNo one_third)) Hm23le0 H0le).
L178438
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r9s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L178441
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) 1 (minus_SNo one_third) Hr9sRx real_1 Hm13R HhiI3).
L178442
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L178444
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L178444
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L178445
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) (minus_SNo one_third)) two_thirds.
L178447
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L178450
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) (minus_SNo one_third)) Hm23R H23R Hr10xR Hlow Hup).
L178454
Assume HxnotC10: ¬ (x ∈ C10).
L178454
We prove the intermediate claim HxX: x ∈ X.
L178456
An exact proof term for the current goal is (HAsubX x HxA).
L178456
We prove the intermediate claim HnotI1: ¬ (apply_fun r9s x ∈ I1).
L178458
Assume Hr9sI1': apply_fun r9s x ∈ I1.
L178458
We prove the intermediate claim Hr9sI1I: apply_fun r9s x ∈ I1 ∩ I.
L178460
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r9s x) Hr9sI1' Hr9sIx).
L178460
We prove the intermediate claim HxB10': x ∈ B10.
L178462
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r9s x0 ∈ I1 ∩ I) x HxA Hr9sI1I).
L178462
Apply FalseE to the current goal.
L178463
An exact proof term for the current goal is (HxnotB10 HxB10').
L178464
We prove the intermediate claim HnotI3: ¬ (apply_fun r9s x ∈ I3).
L178466
Assume Hr9sI3': apply_fun r9s x ∈ I3.
L178466
We prove the intermediate claim Hr9sI3I: apply_fun r9s x ∈ I3 ∩ I.
L178468
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r9s x) Hr9sI3' Hr9sIx).
L178468
We prove the intermediate claim HxC10': x ∈ C10.
L178470
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r9s x0 ∈ I3 ∩ I) x HxA Hr9sI3I).
L178470
Apply FalseE to the current goal.
L178471
An exact proof term for the current goal is (HxnotC10 HxC10').
L178472
We prove the intermediate claim Hr9sRx: apply_fun r9s x ∈ R.
L178474
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r9s x) Hr9sIx).
L178474
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178476
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178476
We prove the intermediate claim H13R: one_third ∈ R.
L178478
An exact proof term for the current goal is one_third_in_R.
L178478
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178480
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178480
We prove the intermediate claim Hr9s_boundsI: Rle (minus_SNo 1) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) 1.
L178482
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r9s x) Hm1R real_1 Hr9sIx).
L178482
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r9s x) (minus_SNo 1)).
L178484
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r9s x) (andEL (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_boundsI)).
L178485
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r9s x)).
L178487
An exact proof term for the current goal is (RleE_nlt (apply_fun r9s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r9s x)) (Rle (apply_fun r9s x) 1) Hr9s_boundsI)).
L178488
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r9s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r9s x).
L178490
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r9s x) Hm1R Hm13R Hr9sRx HnotI1).
L178491
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r9s x).
L178493
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r9s x))) to the current goal.
L178494
Assume Hbad: Rlt (apply_fun r9s x) (minus_SNo 1).
L178494
Apply FalseE to the current goal.
L178495
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L178497
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r9s x).
L178497
An exact proof term for the current goal is Hok.
L178498
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r9s x) (minus_SNo one_third)).
L178500
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r9s x) Hm13lt_fx).
L178500
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r9s x) one_third ∨ Rlt 1 (apply_fun r9s x).
L178502
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r9s x) H13R real_1 Hr9sRx HnotI3).
L178503
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r9s x) one_third.
L178505
Apply (HnotI3_cases (Rlt (apply_fun r9s x) one_third)) to the current goal.
L178506
Assume Hok: Rlt (apply_fun r9s x) one_third.
L178506
An exact proof term for the current goal is Hok.
L178508
Assume Hbad: Rlt 1 (apply_fun r9s x).
L178508
Apply FalseE to the current goal.
L178509
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L178510
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r9s x)).
L178512
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r9s x) one_third Hfx_lt_13).
L178512
We prove the intermediate claim Hr9sI0: apply_fun r9s x ∈ I0.
L178514
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L178515
We prove the intermediate claim HxSep: apply_fun r9s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L178517
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r9s x) Hr9sRx (andI (¬ (Rlt (apply_fun r9s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r9s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L178521
rewrite the current goal using HI0_def (from left to right).
L178522
An exact proof term for the current goal is HxSep.
L178523
We prove the intermediate claim Hu10funI0: function_on u10 X I0.
L178525
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u10 Hu10contI0).
L178525
We prove the intermediate claim Hu10xI0: apply_fun u10 x ∈ I0.
L178527
An exact proof term for the current goal is (Hu10funI0 x HxX).
L178527
We prove the intermediate claim Hu10xR: apply_fun u10 x ∈ R.
L178529
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u10 x) Hu10xI0).
L178529
We prove the intermediate claim Hm_u10x_R: minus_SNo (apply_fun u10 x) ∈ R.
L178531
An exact proof term for the current goal is (real_minus_SNo (apply_fun u10 x) Hu10xR).
L178531
rewrite the current goal using (Hr10_apply x HxA) (from left to right).
L178532
We prove the intermediate claim Hr10xR: add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) ∈ R.
L178534
An exact proof term for the current goal is (real_add_SNo (apply_fun r9s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) (minus_SNo (apply_fun u10 x)) Hm_u10x_R).
L178536
We prove the intermediate claim H23R: two_thirds ∈ R.
L178538
An exact proof term for the current goal is two_thirds_in_R.
L178538
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178540
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178540
We prove the intermediate claim Hr9s_bounds0: Rle (minus_SNo one_third) (apply_fun r9s x) ∧ Rle (apply_fun r9s x) one_third.
L178542
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r9s x) Hm13R H13R Hr9sI0).
L178542
We prove the intermediate claim Hu10_bounds0: Rle (minus_SNo one_third) (apply_fun u10 x) ∧ Rle (apply_fun u10 x) one_third.
L178544
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u10 x) Hm13R H13R Hu10xI0).
L178544
We prove the intermediate claim Hr9s_le_13: Rle (apply_fun r9s x) one_third.
L178546
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r9s x)) (Rle (apply_fun r9s x) one_third) Hr9s_bounds0).
L178549
We prove the intermediate claim Hr9s_ge_m13: Rle (minus_SNo one_third) (apply_fun r9s x).
L178551
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r9s x)) (Rle (apply_fun r9s x) one_third) Hr9s_bounds0).
L178554
We prove the intermediate claim Hu10_le_13: Rle (apply_fun u10 x) one_third.
L178556
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u10 x)) (Rle (apply_fun u10 x) one_third) Hu10_bounds0).
L178559
We prove the intermediate claim Hu10_ge_m13: Rle (minus_SNo one_third) (apply_fun u10 x).
L178561
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u10 x)) (Rle (apply_fun u10 x) one_third) Hu10_bounds0).
L178564
We prove the intermediate claim Hm13_le_mu10: Rle (minus_SNo one_third) (minus_SNo (apply_fun u10 x)).
L178566
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u10 x) one_third Hu10_le_13).
L178566
We prove the intermediate claim Hmu10_le_13: Rle (minus_SNo (apply_fun u10 x)) one_third.
L178568
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u10 x)) (minus_SNo (minus_SNo one_third)).
L178569
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u10 x) Hu10_ge_m13).
L178569
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L178570
An exact proof term for the current goal is Htmp.
L178571
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))).
L178574
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u10 x)) Hm13R Hm13R Hm_u10x_R Hm13_le_mu10).
L178575
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L178578
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) Hm_u10x_R Hr9s_ge_m13).
L178581
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L178584
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) Hlow1 Hlow2).
L178587
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))).
L178589
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L178589
An exact proof term for the current goal is Hlow_tmp.
L178590
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) one_third).
L178593
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r9s x) (minus_SNo (apply_fun u10 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) Hm_u10x_R H13R Hmu10_le_13).
L178595
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r9s x) one_third) (add_SNo one_third one_third).
L178598
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r9s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r9s x) Hr9sI0) H13R H13R Hr9s_le_13).
L178600
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo one_third one_third).
L178603
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) (add_SNo (apply_fun r9s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L178606
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L178608
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) two_thirds.
L178610
rewrite the current goal using Hdef23 (from left to right) at position 1.
L178610
An exact proof term for the current goal is Hup_tmp.
L178611
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r9s x) (minus_SNo (apply_fun u10 x))) Hm23R H23R Hr10xR Hlow Hup).
L178614
Set r10s to be the term compose_fun A r10 (div_const_fun den).
L178615
We prove the intermediate claim Hr10s_cont: continuous_map A Ta I Ti r10s.
L178617
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L178619
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L178619
We prove the intermediate claim Hr10s_contR: continuous_map A Ta R R_standard_topology r10s.
L178621
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r10 (div_const_fun den) Hr10_cont Hdivcont).
L178622
We prove the intermediate claim Hr10s_I: ∀x : set, x ∈ A → apply_fun r10s x ∈ I.
L178624
Let x be given.
L178624
Assume HxA: x ∈ A.
L178624
We prove the intermediate claim Hr10xI2: apply_fun r10 x ∈ I2.
L178626
An exact proof term for the current goal is (Hr10_range x HxA).
L178626
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L178628
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178628
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L178630
An exact proof term for the current goal is (real_minus_SNo den H23R).
L178630
We prove the intermediate claim Hr10xR: apply_fun r10 x ∈ R.
L178632
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r10 x) Hr10xI2).
L178632
We prove the intermediate claim Hr10xS: SNo (apply_fun r10 x).
L178634
An exact proof term for the current goal is (real_SNo (apply_fun r10 x) Hr10xR).
L178634
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r10 x) ∧ Rle (apply_fun r10 x) den.
L178636
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r10 x) HmdenR H23R Hr10xI2).
L178636
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r10 x).
L178638
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r10 x)) (Rle (apply_fun r10 x) den) Hbounds).
L178640
We prove the intermediate claim Hhi: Rle (apply_fun r10 x) den.
L178642
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r10 x)) (Rle (apply_fun r10 x) den) Hbounds).
L178644
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r10 x)).
L178646
An exact proof term for the current goal is (RleE_nlt (apply_fun r10 x) den Hhi).
L178646
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r10 x) (minus_SNo den)).
L178648
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r10 x) Hlo).
L178648
We prove the intermediate claim HyEq: apply_fun r10s x = div_SNo (apply_fun r10 x) den.
L178650
rewrite the current goal using (compose_fun_apply A r10 (div_const_fun den) x HxA) (from left to right).
L178650
rewrite the current goal using (div_const_fun_apply den (apply_fun r10 x) H23R Hr10xR) (from left to right).
Use reflexivity.
L178652
We prove the intermediate claim HyR: apply_fun r10s x ∈ R.
L178654
rewrite the current goal using HyEq (from left to right).
L178654
An exact proof term for the current goal is (real_div_SNo (apply_fun r10 x) Hr10xR den H23R).
L178655
We prove the intermediate claim HyS: SNo (apply_fun r10s x).
L178657
An exact proof term for the current goal is (real_SNo (apply_fun r10s x) HyR).
L178657
We prove the intermediate claim Hy_le_1: Rle (apply_fun r10s x) 1.
L178659
Apply (RleI (apply_fun r10s x) 1 HyR real_1) to the current goal.
L178659
We will prove ¬ (Rlt 1 (apply_fun r10s x)).
L178660
Assume H1lt: Rlt 1 (apply_fun r10s x).
L178661
We prove the intermediate claim H1lty: 1 < apply_fun r10s x.
L178663
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r10s x) H1lt).
L178663
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r10s x) den.
L178665
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r10s x) den SNo_1 HyS H23S H23pos H1lty).
L178665
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r10s x) den.
L178667
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L178667
An exact proof term for the current goal is HmulLt.
L178668
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r10s x) den = apply_fun r10 x.
L178670
rewrite the current goal using HyEq (from left to right) at position 1.
L178670
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r10 x) den Hr10xS H23S H23ne0).
L178671
We prove the intermediate claim Hden_lt_r10x: den < apply_fun r10 x.
L178673
rewrite the current goal using HmulEq (from right to left).
L178673
An exact proof term for the current goal is HmulLt'.
L178674
We prove the intermediate claim Hbad: Rlt den (apply_fun r10 x).
L178676
An exact proof term for the current goal is (RltI den (apply_fun r10 x) H23R Hr10xR Hden_lt_r10x).
L178676
An exact proof term for the current goal is (Hnlt_hi Hbad).
L178677
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r10s x).
L178679
Apply (RleI (minus_SNo 1) (apply_fun r10s x) Hm1R HyR) to the current goal.
L178679
We will prove ¬ (Rlt (apply_fun r10s x) (minus_SNo 1)).
L178680
Assume Hylt: Rlt (apply_fun r10s x) (minus_SNo 1).
L178681
We prove the intermediate claim Hylts: apply_fun r10s x < minus_SNo 1.
L178683
An exact proof term for the current goal is (RltE_lt (apply_fun r10s x) (minus_SNo 1) Hylt).
L178683
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r10s x) den < mul_SNo (minus_SNo 1) den.
L178685
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r10s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L178686
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r10s x) den = apply_fun r10 x.
L178688
rewrite the current goal using HyEq (from left to right) at position 1.
L178688
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r10 x) den Hr10xS H23S H23ne0).
L178689
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L178691
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L178691
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L178693
We prove the intermediate claim Hr10x_lt_mden: apply_fun r10 x < minus_SNo den.
L178695
rewrite the current goal using HmulEq (from right to left).
L178695
rewrite the current goal using HrhsEq (from right to left).
L178696
An exact proof term for the current goal is HmulLt.
L178697
We prove the intermediate claim Hbad: Rlt (apply_fun r10 x) (minus_SNo den).
L178699
An exact proof term for the current goal is (RltI (apply_fun r10 x) (minus_SNo den) Hr10xR HmdenR Hr10x_lt_mden).
L178699
An exact proof term for the current goal is (Hnlt_lo Hbad).
L178700
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r10s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L178702
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r10s I Hr10s_contR HIcR Hr10s_I).
L178704
We prove the intermediate claim Hex_u11: ∃u11 : set, continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x = one_third).
(*** further iteration scaffold: build u11 and g11 from residual r10s ***)
L178714
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r10s Hnorm HA Hr10s_cont).
L178714
Apply Hex_u11 to the current goal.
L178715
Let u11 be given.
L178716
Assume Hu11.
L178716
We prove the intermediate claim Hu11contI0: continuous_map X Tx I0 T0 u11.
L178718
We prove the intermediate claim Hu11AB: continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third).
L178722
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11 ∧ (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x = one_third) Hu11).
L178728
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11) (∀x : set, x ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x = minus_SNo one_third) Hu11AB).
L178733
We prove the intermediate claim Hu11contR: continuous_map X Tx R R_standard_topology u11.
L178735
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L178736
We prove the intermediate claim HI0subR: I0 ⊆ R.
L178738
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L178738
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u11 Hu11contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L178746
Set den11 to be the term mul_SNo den10 den.
L178747
We prove the intermediate claim Hden11R: den11 ∈ R.
L178749
An exact proof term for the current goal is (real_mul_SNo den10 Hden10R den H23R).
L178749
We prove the intermediate claim Hden11pos: 0 < den11.
L178751
We prove the intermediate claim Hden10S: SNo den10.
L178752
An exact proof term for the current goal is (real_SNo den10 Hden10R).
L178752
An exact proof term for the current goal is (mul_SNo_pos_pos den10 den Hden10S H23S Hden10pos HdenPos).
L178753
Set u11s to be the term compose_fun X u11 (mul_const_fun den11).
L178754
We prove the intermediate claim Hu11s_cont: continuous_map X Tx R R_standard_topology u11s.
L178756
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u11 den11 HTx Hu11contR Hden11R Hden11pos).
L178756
Set g11 to be the term compose_fun X (pair_map X g10 u11s) add_fun_R.
L178757
We prove the intermediate claim Hg11cont: continuous_map X Tx R R_standard_topology g11.
L178759
An exact proof term for the current goal is (add_two_continuous_R X Tx g10 u11s HTx Hg10cont Hu11s_cont).
L178759
We prove the intermediate claim Hu11contA: continuous_map A Ta R R_standard_topology u11.
(*** further iteration scaffold: residual r11 on A and scaling r11s ***)
L178763
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u11 A HTx HAsubX Hu11contR).
L178763
Set u11neg to be the term compose_fun A u11 neg_fun.
L178764
We prove the intermediate claim Hu11neg_cont: continuous_map A Ta R R_standard_topology u11neg.
L178766
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u11 neg_fun Hu11contA Hnegcont).
L178767
We prove the intermediate claim Hr10s_contR: continuous_map A Ta R R_standard_topology r10s.
L178769
We prove the intermediate claim HTiEq'': Ti = subspace_topology R R_standard_topology I.
L178770
An exact proof term for the current goal is HTiEq.
L178770
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r10s Hr10s_cont HIcR R_standard_topology_is_topology_local HTiEq'').
L178772
Set r11 to be the term compose_fun A (pair_map A r10s u11neg) add_fun_R.
L178773
We prove the intermediate claim Hr11_cont: continuous_map A Ta R R_standard_topology r11.
L178775
An exact proof term for the current goal is (add_two_continuous_R A Ta r10s u11neg HTa Hr10s_contR Hu11neg_cont).
L178775
We prove the intermediate claim Hr11_apply: ∀x : set, x ∈ A → apply_fun r11 x = add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x)).
L178778
Let x be given.
L178778
Assume HxA: x ∈ A.
L178778
We prove the intermediate claim Hpimg: apply_fun (pair_map A r10s u11neg) x ∈ setprod R R.
L178780
rewrite the current goal using (pair_map_apply A R R r10s u11neg x HxA) (from left to right).
L178780
We prove the intermediate claim Hr10sxI: apply_fun r10s x ∈ I.
L178782
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r10s Hr10s_cont x HxA).
L178782
We prove the intermediate claim Hr10sxR: apply_fun r10s x ∈ R.
L178784
An exact proof term for the current goal is (HIcR (apply_fun r10s x) Hr10sxI).
L178784
We prove the intermediate claim Hu11negRx: apply_fun u11neg x ∈ R.
L178786
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u11neg Hu11neg_cont x HxA).
L178786
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r10s x) (apply_fun u11neg x) Hr10sxR Hu11negRx).
L178787
rewrite the current goal using (compose_fun_apply A (pair_map A r10s u11neg) add_fun_R x HxA) (from left to right).
L178788
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r10s u11neg) x) Hpimg) (from left to right) at position 1.
L178789
rewrite the current goal using (pair_map_apply A R R r10s u11neg x HxA) (from left to right).
L178790
rewrite the current goal using (tuple_2_0_eq (apply_fun r10s x) (apply_fun u11neg x)) (from left to right).
L178791
rewrite the current goal using (tuple_2_1_eq (apply_fun r10s x) (apply_fun u11neg x)) (from left to right).
L178792
rewrite the current goal using (compose_fun_apply A u11 neg_fun x HxA) (from left to right) at position 1.
L178793
We prove the intermediate claim Hu11Rx: apply_fun u11 x ∈ R.
L178795
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u11 Hu11contA x HxA).
L178795
rewrite the current goal using (neg_fun_apply (apply_fun u11 x) Hu11Rx) (from left to right) at position 1.
Use reflexivity.
L178797
We prove the intermediate claim Hr11_range: ∀x : set, x ∈ A → apply_fun r11 x ∈ I2.
L178799
We prove the intermediate claim Hu11AB: continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third).
L178803
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third) Hu11).
L178809
We prove the intermediate claim Hu11_on_B11: ∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third.
L178813
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u11) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third) Hu11AB).
L178817
We prove the intermediate claim Hu11_on_C11: ∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third.
L178821
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u11 ∧ (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u11 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r10s ((closed_interval one_third 1) ∩ I) → apply_fun u11 x0 = one_third) Hu11).
L178827
Let x be given.
L178828
Assume HxA: x ∈ A.
L178828
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L178829
Set I3 to be the term closed_interval one_third 1.
L178830
Set B11 to be the term preimage_of A r10s (I1 ∩ I).
L178831
Set C11 to be the term preimage_of A r10s (I3 ∩ I).
L178832
We prove the intermediate claim Hr10sIx: apply_fun r10s x ∈ I.
L178834
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r10s Hr10s_cont x HxA).
L178834
We prove the intermediate claim HB11_cases: x ∈ B11 ∨ ¬ (x ∈ B11).
L178836
An exact proof term for the current goal is (xm (x ∈ B11)).
L178836
Apply (HB11_cases (apply_fun r11 x ∈ I2)) to the current goal.
L178838
Assume HxB11: x ∈ B11.
L178838
We prove the intermediate claim Hu11eq: apply_fun u11 x = minus_SNo one_third.
L178840
An exact proof term for the current goal is (Hu11_on_B11 x HxB11).
L178840
We prove the intermediate claim Hr11eq: apply_fun r11 x = add_SNo (apply_fun r10s x) one_third.
L178842
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L178842
rewrite the current goal using Hu11eq (from left to right) at position 1.
L178843
We prove the intermediate claim H13R: one_third ∈ R.
L178845
An exact proof term for the current goal is one_third_in_R.
L178845
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L178847
rewrite the current goal using Hr11eq (from left to right).
L178848
We prove the intermediate claim Hr10sI1I: apply_fun r10s x ∈ I1 ∩ I.
L178850
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r10s x0 ∈ I1 ∩ I) x HxB11).
L178850
We prove the intermediate claim Hr10sI1: apply_fun r10s x ∈ I1.
L178852
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r10s x) Hr10sI1I).
L178852
We prove the intermediate claim H13R: one_third ∈ R.
L178854
An exact proof term for the current goal is one_third_in_R.
L178854
We prove the intermediate claim H23R: two_thirds ∈ R.
L178856
An exact proof term for the current goal is two_thirds_in_R.
L178856
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178858
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178858
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178860
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178860
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178862
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178862
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L178864
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hr10sI1).
L178864
We prove the intermediate claim Hr10s_bounds: Rle (minus_SNo 1) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) (minus_SNo one_third).
L178866
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hm1R Hm13R Hr10sI1).
L178867
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r10s x).
L178869
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) (minus_SNo one_third)) Hr10s_bounds).
L178870
We prove the intermediate claim HhiI1: Rle (apply_fun r10s x) (minus_SNo one_third).
L178872
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) (minus_SNo one_third)) Hr10s_bounds).
L178873
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) one_third ∈ R.
L178875
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) Hr10sRx one_third H13R).
L178875
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r10s x) one_third).
L178877
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r10s x) one_third Hm1R Hr10sRx H13R Hm1le).
L178877
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) one_third).
L178879
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L178879
An exact proof term for the current goal is Hlow_tmp.
L178880
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r10s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L178882
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) (minus_SNo one_third) one_third Hr10sRx Hm13R H13R HhiI1).
L178882
We prove the intermediate claim H13S: SNo one_third.
L178884
An exact proof term for the current goal is (real_SNo one_third H13R).
L178884
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r10s x) one_third) 0.
L178886
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L178886
An exact proof term for the current goal is Hup0_tmp.
L178887
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L178889
An exact proof term for the current goal is Rle_0_two_thirds.
L178889
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) one_third) two_thirds.
L178891
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) one_third) 0 two_thirds Hup0 H0le23).
L178891
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) one_third) Hm23R H23R Hr11xR Hlow Hup).
L178895
Assume HxnotB11: ¬ (x ∈ B11).
L178895
We prove the intermediate claim HC11_cases: x ∈ C11 ∨ ¬ (x ∈ C11).
L178897
An exact proof term for the current goal is (xm (x ∈ C11)).
L178897
Apply (HC11_cases (apply_fun r11 x ∈ I2)) to the current goal.
L178899
Assume HxC11: x ∈ C11.
L178899
We prove the intermediate claim Hu11eq: apply_fun u11 x = one_third.
L178901
An exact proof term for the current goal is (Hu11_on_C11 x HxC11).
L178901
We prove the intermediate claim Hr11eq: apply_fun r11 x = add_SNo (apply_fun r10s x) (minus_SNo one_third).
L178903
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L178903
rewrite the current goal using Hu11eq (from left to right) at position 1.
Use reflexivity.
L178905
rewrite the current goal using Hr11eq (from left to right).
L178906
We prove the intermediate claim Hr10sI3I: apply_fun r10s x ∈ I3 ∩ I.
L178908
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r10s x0 ∈ I3 ∩ I) x HxC11).
L178908
We prove the intermediate claim Hr10sI3: apply_fun r10s x ∈ I3.
L178910
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r10s x) Hr10sI3I).
L178910
We prove the intermediate claim H13R: one_third ∈ R.
L178912
An exact proof term for the current goal is one_third_in_R.
L178912
We prove the intermediate claim H23R: two_thirds ∈ R.
L178914
An exact proof term for the current goal is two_thirds_in_R.
L178914
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L178916
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L178916
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178918
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178918
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L178920
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r10s x) Hr10sI3).
L178920
We prove the intermediate claim Hr10s_bounds: Rle one_third (apply_fun r10s x) ∧ Rle (apply_fun r10s x) 1.
L178922
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r10s x) H13R real_1 Hr10sI3).
L178922
We prove the intermediate claim HloI3: Rle one_third (apply_fun r10s x).
L178924
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_bounds).
L178924
We prove the intermediate claim HhiI3: Rle (apply_fun r10s x) 1.
L178926
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_bounds).
L178926
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) (minus_SNo one_third) ∈ R.
L178928
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) Hr10sRx (minus_SNo one_third) Hm13R).
L178928
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L178931
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r10s x) (minus_SNo one_third) H13R Hr10sRx Hm13R HloI3).
L178932
We prove the intermediate claim H13S: SNo one_third.
L178934
An exact proof term for the current goal is (real_SNo one_third H13R).
L178934
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L178936
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L178936
An exact proof term for the current goal is H0le_tmp.
L178937
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L178939
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L178939
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) (minus_SNo one_third)).
L178941
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r10s x) (minus_SNo one_third)) Hm23le0 H0le).
L178943
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r10s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L178946
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) 1 (minus_SNo one_third) Hr10sRx real_1 Hm13R HhiI3).
L178947
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L178949
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L178949
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L178950
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) (minus_SNo one_third)) two_thirds.
L178952
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L178955
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) (minus_SNo one_third)) Hm23R H23R Hr11xR Hlow Hup).
L178959
Assume HxnotC11: ¬ (x ∈ C11).
L178959
We prove the intermediate claim HxX: x ∈ X.
L178961
An exact proof term for the current goal is (HAsubX x HxA).
L178961
We prove the intermediate claim HnotI1: ¬ (apply_fun r10s x ∈ I1).
L178963
Assume Hr10sI1': apply_fun r10s x ∈ I1.
L178963
We prove the intermediate claim Hr10sI1I: apply_fun r10s x ∈ I1 ∩ I.
L178965
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r10s x) Hr10sI1' Hr10sIx).
L178965
We prove the intermediate claim HxB11': x ∈ B11.
L178967
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r10s x0 ∈ I1 ∩ I) x HxA Hr10sI1I).
L178967
Apply FalseE to the current goal.
L178968
An exact proof term for the current goal is (HxnotB11 HxB11').
L178969
We prove the intermediate claim HnotI3: ¬ (apply_fun r10s x ∈ I3).
L178971
Assume Hr10sI3': apply_fun r10s x ∈ I3.
L178971
We prove the intermediate claim Hr10sI3I: apply_fun r10s x ∈ I3 ∩ I.
L178973
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r10s x) Hr10sI3' Hr10sIx).
L178973
We prove the intermediate claim HxC11': x ∈ C11.
L178975
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r10s x0 ∈ I3 ∩ I) x HxA Hr10sI3I).
L178975
Apply FalseE to the current goal.
L178976
An exact proof term for the current goal is (HxnotC11 HxC11').
L178977
We prove the intermediate claim Hr10sRx: apply_fun r10s x ∈ R.
L178979
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r10s x) Hr10sIx).
L178979
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L178981
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L178981
We prove the intermediate claim H13R: one_third ∈ R.
L178983
An exact proof term for the current goal is one_third_in_R.
L178983
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L178985
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L178985
We prove the intermediate claim Hr10s_boundsI: Rle (minus_SNo 1) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) 1.
L178987
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r10s x) Hm1R real_1 Hr10sIx).
L178987
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r10s x) (minus_SNo 1)).
L178989
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r10s x) (andEL (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_boundsI)).
L178990
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r10s x)).
L178992
An exact proof term for the current goal is (RleE_nlt (apply_fun r10s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r10s x)) (Rle (apply_fun r10s x) 1) Hr10s_boundsI)).
L178993
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r10s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r10s x).
L178995
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r10s x) Hm1R Hm13R Hr10sRx HnotI1).
L178996
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r10s x).
L178998
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r10s x))) to the current goal.
L178999
Assume Hbad: Rlt (apply_fun r10s x) (minus_SNo 1).
L178999
Apply FalseE to the current goal.
L179000
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L179002
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r10s x).
L179002
An exact proof term for the current goal is Hok.
L179003
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r10s x) (minus_SNo one_third)).
L179005
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r10s x) Hm13lt_fx).
L179005
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r10s x) one_third ∨ Rlt 1 (apply_fun r10s x).
L179007
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r10s x) H13R real_1 Hr10sRx HnotI3).
L179008
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r10s x) one_third.
L179010
Apply (HnotI3_cases (Rlt (apply_fun r10s x) one_third)) to the current goal.
L179011
Assume Hok: Rlt (apply_fun r10s x) one_third.
L179011
An exact proof term for the current goal is Hok.
L179013
Assume Hbad: Rlt 1 (apply_fun r10s x).
L179013
Apply FalseE to the current goal.
L179014
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L179015
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r10s x)).
L179017
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r10s x) one_third Hfx_lt_13).
L179017
We prove the intermediate claim Hr10sI0: apply_fun r10s x ∈ I0.
L179019
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L179020
We prove the intermediate claim HxSep: apply_fun r10s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L179022
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r10s x) Hr10sRx (andI (¬ (Rlt (apply_fun r10s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r10s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L179026
rewrite the current goal using HI0_def (from left to right).
L179027
An exact proof term for the current goal is HxSep.
L179028
We prove the intermediate claim Hu11funI0: function_on u11 X I0.
L179030
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u11 Hu11contI0).
L179030
We prove the intermediate claim Hu11xI0: apply_fun u11 x ∈ I0.
L179032
An exact proof term for the current goal is (Hu11funI0 x HxX).
L179032
We prove the intermediate claim Hu11xR: apply_fun u11 x ∈ R.
L179034
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u11 x) Hu11xI0).
L179034
We prove the intermediate claim Hm_u11x_R: minus_SNo (apply_fun u11 x) ∈ R.
L179036
An exact proof term for the current goal is (real_minus_SNo (apply_fun u11 x) Hu11xR).
L179036
rewrite the current goal using (Hr11_apply x HxA) (from left to right).
L179037
We prove the intermediate claim Hr11xR: add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) ∈ R.
L179039
An exact proof term for the current goal is (real_add_SNo (apply_fun r10s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) (minus_SNo (apply_fun u11 x)) Hm_u11x_R).
L179041
We prove the intermediate claim H23R: two_thirds ∈ R.
L179043
An exact proof term for the current goal is two_thirds_in_R.
L179043
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179045
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179045
We prove the intermediate claim Hr10s_bounds0: Rle (minus_SNo one_third) (apply_fun r10s x) ∧ Rle (apply_fun r10s x) one_third.
L179047
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r10s x) Hm13R H13R Hr10sI0).
L179047
We prove the intermediate claim Hu11_bounds0: Rle (minus_SNo one_third) (apply_fun u11 x) ∧ Rle (apply_fun u11 x) one_third.
L179049
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u11 x) Hm13R H13R Hu11xI0).
L179049
We prove the intermediate claim Hr10s_le_13: Rle (apply_fun r10s x) one_third.
L179051
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r10s x)) (Rle (apply_fun r10s x) one_third) Hr10s_bounds0).
L179054
We prove the intermediate claim Hr10s_ge_m13: Rle (minus_SNo one_third) (apply_fun r10s x).
L179056
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r10s x)) (Rle (apply_fun r10s x) one_third) Hr10s_bounds0).
L179059
We prove the intermediate claim Hu11_le_13: Rle (apply_fun u11 x) one_third.
L179061
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u11 x)) (Rle (apply_fun u11 x) one_third) Hu11_bounds0).
L179064
We prove the intermediate claim Hu11_ge_m13: Rle (minus_SNo one_third) (apply_fun u11 x).
L179066
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u11 x)) (Rle (apply_fun u11 x) one_third) Hu11_bounds0).
L179069
We prove the intermediate claim Hm13_le_mu11: Rle (minus_SNo one_third) (minus_SNo (apply_fun u11 x)).
L179071
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u11 x) one_third Hu11_le_13).
L179071
We prove the intermediate claim Hmu11_le_13: Rle (minus_SNo (apply_fun u11 x)) one_third.
L179073
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u11 x)) (minus_SNo (minus_SNo one_third)).
L179074
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u11 x) Hu11_ge_m13).
L179074
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L179075
An exact proof term for the current goal is Htmp.
L179076
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))).
L179079
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u11 x)) Hm13R Hm13R Hm_u11x_R Hm13_le_mu11).
L179080
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L179083
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) Hm_u11x_R Hr10s_ge_m13).
L179086
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L179089
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) Hlow1 Hlow2).
L179092
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))).
L179094
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L179094
An exact proof term for the current goal is Hlow_tmp.
L179095
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) one_third).
L179098
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r10s x) (minus_SNo (apply_fun u11 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) Hm_u11x_R H13R Hmu11_le_13).
L179100
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r10s x) one_third) (add_SNo one_third one_third).
L179103
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r10s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r10s x) Hr10sI0) H13R H13R Hr10s_le_13).
L179105
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo one_third one_third).
L179108
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) (add_SNo (apply_fun r10s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L179111
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L179113
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) two_thirds.
L179115
rewrite the current goal using Hdef23 (from left to right) at position 1.
L179115
An exact proof term for the current goal is Hup_tmp.
L179116
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r10s x) (minus_SNo (apply_fun u11 x))) Hm23R H23R Hr11xR Hlow Hup).
L179119
Set r11s to be the term compose_fun A r11 (div_const_fun den).
L179120
We prove the intermediate claim Hr11s_cont: continuous_map A Ta I Ti r11s.
L179122
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L179124
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L179124
We prove the intermediate claim Hr11s_contR: continuous_map A Ta R R_standard_topology r11s.
L179126
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r11 (div_const_fun den) Hr11_cont Hdivcont).
L179127
We prove the intermediate claim Hr11s_I: ∀x : set, x ∈ A → apply_fun r11s x ∈ I.
L179129
Let x be given.
L179129
Assume HxA: x ∈ A.
L179129
We prove the intermediate claim Hr11xI2: apply_fun r11 x ∈ I2.
L179131
An exact proof term for the current goal is (Hr11_range x HxA).
L179131
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L179133
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179133
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L179135
An exact proof term for the current goal is (real_minus_SNo den H23R).
L179135
We prove the intermediate claim Hr11xR: apply_fun r11 x ∈ R.
L179137
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r11 x) Hr11xI2).
L179137
We prove the intermediate claim Hr11xS: SNo (apply_fun r11 x).
L179139
An exact proof term for the current goal is (real_SNo (apply_fun r11 x) Hr11xR).
L179139
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r11 x) ∧ Rle (apply_fun r11 x) den.
L179141
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r11 x) HmdenR H23R Hr11xI2).
L179141
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r11 x).
L179143
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r11 x)) (Rle (apply_fun r11 x) den) Hbounds).
L179145
We prove the intermediate claim Hhi: Rle (apply_fun r11 x) den.
L179147
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r11 x)) (Rle (apply_fun r11 x) den) Hbounds).
L179149
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r11 x)).
L179151
An exact proof term for the current goal is (RleE_nlt (apply_fun r11 x) den Hhi).
L179151
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r11 x) (minus_SNo den)).
L179153
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r11 x) Hlo).
L179153
We prove the intermediate claim HyEq: apply_fun r11s x = div_SNo (apply_fun r11 x) den.
L179155
rewrite the current goal using (compose_fun_apply A r11 (div_const_fun den) x HxA) (from left to right).
L179155
rewrite the current goal using (div_const_fun_apply den (apply_fun r11 x) H23R Hr11xR) (from left to right).
Use reflexivity.
L179157
We prove the intermediate claim HyR: apply_fun r11s x ∈ R.
L179159
rewrite the current goal using HyEq (from left to right).
L179159
An exact proof term for the current goal is (real_div_SNo (apply_fun r11 x) Hr11xR den H23R).
L179160
We prove the intermediate claim HyS: SNo (apply_fun r11s x).
L179162
An exact proof term for the current goal is (real_SNo (apply_fun r11s x) HyR).
L179162
We prove the intermediate claim Hy_le_1: Rle (apply_fun r11s x) 1.
L179164
Apply (RleI (apply_fun r11s x) 1 HyR real_1) to the current goal.
L179164
We will prove ¬ (Rlt 1 (apply_fun r11s x)).
L179165
Assume H1lt: Rlt 1 (apply_fun r11s x).
L179166
We prove the intermediate claim H1lty: 1 < apply_fun r11s x.
L179168
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r11s x) H1lt).
L179168
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r11s x) den.
L179170
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r11s x) den SNo_1 HyS H23S H23pos H1lty).
L179170
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r11s x) den.
L179172
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L179172
An exact proof term for the current goal is HmulLt.
L179173
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r11s x) den = apply_fun r11 x.
L179175
rewrite the current goal using HyEq (from left to right) at position 1.
L179175
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r11 x) den Hr11xS H23S H23ne0).
L179176
We prove the intermediate claim Hden_lt_r11x: den < apply_fun r11 x.
L179178
rewrite the current goal using HmulEq (from right to left).
L179178
An exact proof term for the current goal is HmulLt'.
L179179
We prove the intermediate claim Hbad: Rlt den (apply_fun r11 x).
L179181
An exact proof term for the current goal is (RltI den (apply_fun r11 x) H23R Hr11xR Hden_lt_r11x).
L179181
An exact proof term for the current goal is (Hnlt_hi Hbad).
L179182
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r11s x).
L179184
Apply (RleI (minus_SNo 1) (apply_fun r11s x) Hm1R HyR) to the current goal.
L179184
We will prove ¬ (Rlt (apply_fun r11s x) (minus_SNo 1)).
L179185
Assume Hylt: Rlt (apply_fun r11s x) (minus_SNo 1).
L179186
We prove the intermediate claim Hylts: apply_fun r11s x < minus_SNo 1.
L179188
An exact proof term for the current goal is (RltE_lt (apply_fun r11s x) (minus_SNo 1) Hylt).
L179188
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r11s x) den < mul_SNo (minus_SNo 1) den.
L179190
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r11s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L179191
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r11s x) den = apply_fun r11 x.
L179193
rewrite the current goal using HyEq (from left to right) at position 1.
L179193
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r11 x) den Hr11xS H23S H23ne0).
L179194
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L179196
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L179196
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L179198
We prove the intermediate claim Hr11x_lt_mden: apply_fun r11 x < minus_SNo den.
L179200
rewrite the current goal using HmulEq (from right to left).
L179200
rewrite the current goal using HrhsEq (from right to left).
L179201
An exact proof term for the current goal is HmulLt.
L179202
We prove the intermediate claim Hbad: Rlt (apply_fun r11 x) (minus_SNo den).
L179204
An exact proof term for the current goal is (RltI (apply_fun r11 x) (minus_SNo den) Hr11xR HmdenR Hr11x_lt_mden).
L179204
An exact proof term for the current goal is (Hnlt_lo Hbad).
L179205
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r11s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L179207
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r11s I Hr11s_contR HIcR Hr11s_I).
L179209
We prove the intermediate claim Hex_u12: ∃u12 : set, continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x = one_third).
(*** further iteration scaffold: build u12 and g12 from residual r11s ***)
L179219
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r11s Hnorm HA Hr11s_cont).
L179219
Apply Hex_u12 to the current goal.
L179220
Let u12 be given.
L179221
Assume Hu12.
L179221
We prove the intermediate claim Hu12contI0: continuous_map X Tx I0 T0 u12.
L179223
We prove the intermediate claim Hu12AB: continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third).
L179227
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12 ∧ (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x = one_third) Hu12).
L179233
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12) (∀x : set, x ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x = minus_SNo one_third) Hu12AB).
L179238
We prove the intermediate claim Hu12contR: continuous_map X Tx R R_standard_topology u12.
L179240
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L179241
We prove the intermediate claim HI0subR: I0 ⊆ R.
L179243
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L179243
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology u12 Hu12contI0 HI0subR R_standard_topology_is_topology_local HT0eq).
L179251
Set den12 to be the term mul_SNo den11 den.
L179252
We prove the intermediate claim Hden12R: den12 ∈ R.
L179254
An exact proof term for the current goal is (real_mul_SNo den11 Hden11R den H23R).
L179254
We prove the intermediate claim Hden12pos: 0 < den12.
L179256
We prove the intermediate claim Hden11S: SNo den11.
L179257
An exact proof term for the current goal is (real_SNo den11 Hden11R).
L179257
An exact proof term for the current goal is (mul_SNo_pos_pos den11 den Hden11S H23S Hden11pos HdenPos).
L179258
Set u12s to be the term compose_fun X u12 (mul_const_fun den12).
L179259
We prove the intermediate claim Hu12s_cont: continuous_map X Tx R R_standard_topology u12s.
L179261
An exact proof term for the current goal is (scale_continuous_mul_const_pos X Tx u12 den12 HTx Hu12contR Hden12R Hden12pos).
L179261
Set g12 to be the term compose_fun X (pair_map X g11 u12s) add_fun_R.
L179262
We prove the intermediate claim Hg12cont: continuous_map X Tx R R_standard_topology g12.
L179264
An exact proof term for the current goal is (add_two_continuous_R X Tx g11 u12s HTx Hg11cont Hu12s_cont).
L179264
We prove the intermediate claim Hu12contA: continuous_map A Ta R R_standard_topology u12.
(*** further iteration scaffold: residual r12 on A and scaling r12s ***)
L179268
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology u12 A HTx HAsubX Hu12contR).
L179268
Set u12neg to be the term compose_fun A u12 neg_fun.
L179269
We prove the intermediate claim Hu12neg_cont: continuous_map A Ta R R_standard_topology u12neg.
L179271
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology u12 neg_fun Hu12contA Hnegcont).
L179272
We prove the intermediate claim Hr11s_contR: continuous_map A Ta R R_standard_topology r11s.
L179274
We prove the intermediate claim HTiEq''': Ti = subspace_topology R R_standard_topology I.
L179275
An exact proof term for the current goal is HTiEq.
L179275
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r11s Hr11s_cont HIcR R_standard_topology_is_topology_local HTiEq''').
L179277
Set r12 to be the term compose_fun A (pair_map A r11s u12neg) add_fun_R.
L179278
We prove the intermediate claim Hr12_cont: continuous_map A Ta R R_standard_topology r12.
L179280
An exact proof term for the current goal is (add_two_continuous_R A Ta r11s u12neg HTa Hr11s_contR Hu12neg_cont).
L179280
We prove the intermediate claim Hr12_apply: ∀x : set, x ∈ A → apply_fun r12 x = add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x)).
L179283
Let x be given.
L179283
Assume HxA: x ∈ A.
L179283
We prove the intermediate claim Hpimg: apply_fun (pair_map A r11s u12neg) x ∈ setprod R R.
L179285
rewrite the current goal using (pair_map_apply A R R r11s u12neg x HxA) (from left to right).
L179285
We prove the intermediate claim Hr11sxI: apply_fun r11s x ∈ I.
L179287
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r11s Hr11s_cont x HxA).
L179287
We prove the intermediate claim Hr11sxR: apply_fun r11s x ∈ R.
L179289
An exact proof term for the current goal is (HIcR (apply_fun r11s x) Hr11sxI).
L179289
We prove the intermediate claim Hu12negRx: apply_fun u12neg x ∈ R.
L179291
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u12neg Hu12neg_cont x HxA).
L179291
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun r11s x) (apply_fun u12neg x) Hr11sxR Hu12negRx).
L179292
rewrite the current goal using (compose_fun_apply A (pair_map A r11s u12neg) add_fun_R x HxA) (from left to right).
L179293
rewrite the current goal using (add_fun_R_apply (apply_fun (pair_map A r11s u12neg) x) Hpimg) (from left to right) at position 1.
L179294
rewrite the current goal using (pair_map_apply A R R r11s u12neg x HxA) (from left to right).
L179295
rewrite the current goal using (tuple_2_0_eq (apply_fun r11s x) (apply_fun u12neg x)) (from left to right).
L179296
rewrite the current goal using (tuple_2_1_eq (apply_fun r11s x) (apply_fun u12neg x)) (from left to right).
L179297
rewrite the current goal using (compose_fun_apply A u12 neg_fun x HxA) (from left to right) at position 1.
L179298
We prove the intermediate claim Hu12Rx: apply_fun u12 x ∈ R.
L179300
An exact proof term for the current goal is (continuous_map_function_on A Ta R R_standard_topology u12 Hu12contA x HxA).
L179300
rewrite the current goal using (neg_fun_apply (apply_fun u12 x) Hu12Rx) (from left to right) at position 1.
Use reflexivity.
L179302
We prove the intermediate claim Hr12_range: ∀x : set, x ∈ A → apply_fun r12 x ∈ I2.
L179304
We prove the intermediate claim Hu12AB: continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third).
L179308
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third) Hu12).
L179314
We prove the intermediate claim Hu12_on_B12: ∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third.
L179318
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u12) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third) Hu12AB).
L179322
We prove the intermediate claim Hu12_on_C12: ∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third.
L179326
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 u12 ∧ (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u12 x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r11s ((closed_interval one_third 1) ∩ I) → apply_fun u12 x0 = one_third) Hu12).
L179332
Let x be given.
L179333
Assume HxA: x ∈ A.
L179333
Set I1 to be the term closed_interval (minus_SNo 1) (minus_SNo one_third).
L179334
Set I3 to be the term closed_interval one_third 1.
L179335
Set B12 to be the term preimage_of A r11s (I1 ∩ I).
L179336
Set C12 to be the term preimage_of A r11s (I3 ∩ I).
L179337
We prove the intermediate claim Hr11sIx: apply_fun r11s x ∈ I.
L179339
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r11s Hr11s_cont x HxA).
L179339
We prove the intermediate claim HB12_cases: x ∈ B12 ∨ ¬ (x ∈ B12).
L179341
An exact proof term for the current goal is (xm (x ∈ B12)).
L179341
Apply (HB12_cases (apply_fun r12 x ∈ I2)) to the current goal.
L179343
Assume HxB12: x ∈ B12.
L179343
We prove the intermediate claim Hu12eq: apply_fun u12 x = minus_SNo one_third.
L179345
An exact proof term for the current goal is (Hu12_on_B12 x HxB12).
L179345
We prove the intermediate claim Hr12eq: apply_fun r12 x = add_SNo (apply_fun r11s x) one_third.
L179347
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L179347
rewrite the current goal using Hu12eq (from left to right) at position 1.
L179348
We prove the intermediate claim H13R: one_third ∈ R.
L179350
An exact proof term for the current goal is one_third_in_R.
L179350
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from left to right) at position 1.
Use reflexivity.
L179352
rewrite the current goal using Hr12eq (from left to right).
L179353
We prove the intermediate claim Hr11sI1I: apply_fun r11s x ∈ I1 ∩ I.
L179355
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r11s x0 ∈ I1 ∩ I) x HxB12).
L179355
We prove the intermediate claim Hr11sI1: apply_fun r11s x ∈ I1.
L179357
An exact proof term for the current goal is (binintersectE1 I1 I (apply_fun r11s x) Hr11sI1I).
L179357
We prove the intermediate claim H13R: one_third ∈ R.
L179359
An exact proof term for the current goal is one_third_in_R.
L179359
We prove the intermediate claim H23R: two_thirds ∈ R.
L179361
An exact proof term for the current goal is two_thirds_in_R.
L179361
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179363
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179363
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179365
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179365
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179367
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179367
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L179369
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hr11sI1).
L179369
We prove the intermediate claim Hr11s_bounds: Rle (minus_SNo 1) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) (minus_SNo one_third).
L179371
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hm1R Hm13R Hr11sI1).
L179372
We prove the intermediate claim Hm1le: Rle (minus_SNo 1) (apply_fun r11s x).
L179374
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) (minus_SNo one_third)) Hr11s_bounds).
L179375
We prove the intermediate claim HhiI1: Rle (apply_fun r11s x) (minus_SNo one_third).
L179377
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) (minus_SNo one_third)) Hr11s_bounds).
L179378
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) one_third ∈ R.
L179380
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) Hr11sRx one_third H13R).
L179380
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r11s x) one_third).
L179382
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r11s x) one_third Hm1R Hr11sRx H13R Hm1le).
L179382
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) one_third).
L179384
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L179384
An exact proof term for the current goal is Hlow_tmp.
L179385
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r11s x) one_third) (add_SNo (minus_SNo one_third) one_third).
L179387
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) (minus_SNo one_third) one_third Hr11sRx Hm13R H13R HhiI1).
L179387
We prove the intermediate claim H13S: SNo one_third.
L179389
An exact proof term for the current goal is (real_SNo one_third H13R).
L179389
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r11s x) one_third) 0.
L179391
rewrite the current goal using (add_SNo_minus_SNo_linv one_third H13S) (from right to left) at position 1.
L179391
An exact proof term for the current goal is Hup0_tmp.
L179392
We prove the intermediate claim H0le23: Rle 0 two_thirds.
L179394
An exact proof term for the current goal is Rle_0_two_thirds.
L179394
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) one_third) two_thirds.
L179396
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) one_third) 0 two_thirds Hup0 H0le23).
L179396
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) one_third) Hm23R H23R Hr12xR Hlow Hup).
L179400
Assume HxnotB12: ¬ (x ∈ B12).
L179400
We prove the intermediate claim HC12_cases: x ∈ C12 ∨ ¬ (x ∈ C12).
L179402
An exact proof term for the current goal is (xm (x ∈ C12)).
L179402
Apply (HC12_cases (apply_fun r12 x ∈ I2)) to the current goal.
L179404
Assume HxC12: x ∈ C12.
L179404
We prove the intermediate claim Hu12eq: apply_fun u12 x = one_third.
L179406
An exact proof term for the current goal is (Hu12_on_C12 x HxC12).
L179406
We prove the intermediate claim Hr12eq: apply_fun r12 x = add_SNo (apply_fun r11s x) (minus_SNo one_third).
L179408
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L179408
rewrite the current goal using Hu12eq (from left to right) at position 1.
Use reflexivity.
L179410
rewrite the current goal using Hr12eq (from left to right).
L179411
We prove the intermediate claim Hr11sI3I: apply_fun r11s x ∈ I3 ∩ I.
L179413
An exact proof term for the current goal is (SepE2 A (λx0 : set ⇒ apply_fun r11s x0 ∈ I3 ∩ I) x HxC12).
L179413
We prove the intermediate claim Hr11sI3: apply_fun r11s x ∈ I3.
L179415
An exact proof term for the current goal is (binintersectE1 I3 I (apply_fun r11s x) Hr11sI3I).
L179415
We prove the intermediate claim H13R: one_third ∈ R.
L179417
An exact proof term for the current goal is one_third_in_R.
L179417
We prove the intermediate claim H23R: two_thirds ∈ R.
L179419
An exact proof term for the current goal is two_thirds_in_R.
L179419
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179421
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179421
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179423
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179423
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L179425
An exact proof term for the current goal is (closed_interval_sub_R one_third 1 (apply_fun r11s x) Hr11sI3).
L179425
We prove the intermediate claim Hr11s_bounds: Rle one_third (apply_fun r11s x) ∧ Rle (apply_fun r11s x) 1.
L179427
An exact proof term for the current goal is (closed_interval_bounds one_third 1 (apply_fun r11s x) H13R real_1 Hr11sI3).
L179427
We prove the intermediate claim HloI3: Rle one_third (apply_fun r11s x).
L179429
An exact proof term for the current goal is (andEL (Rle one_third (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_bounds).
L179429
We prove the intermediate claim HhiI3: Rle (apply_fun r11s x) 1.
L179431
An exact proof term for the current goal is (andER (Rle one_third (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_bounds).
L179431
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) (minus_SNo one_third) ∈ R.
L179433
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) Hr11sRx (minus_SNo one_third) Hm13R).
L179433
We prove the intermediate claim H0le_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L179436
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r11s x) (minus_SNo one_third) H13R Hr11sRx Hm13R HloI3).
L179437
We prove the intermediate claim H13S: SNo one_third.
L179439
An exact proof term for the current goal is (real_SNo one_third H13R).
L179439
We prove the intermediate claim H0le: Rle 0 (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L179441
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L179441
An exact proof term for the current goal is H0le_tmp.
L179442
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L179444
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L179444
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) (minus_SNo one_third)).
L179446
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r11s x) (minus_SNo one_third)) Hm23le0 H0le).
L179448
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r11s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L179451
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) 1 (minus_SNo one_third) Hr11sRx real_1 Hm13R HhiI3).
L179452
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L179454
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L179454
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L179455
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) (minus_SNo one_third)) two_thirds.
L179457
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L179460
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) (minus_SNo one_third)) Hm23R H23R Hr12xR Hlow Hup).
L179464
Assume HxnotC12: ¬ (x ∈ C12).
L179464
We prove the intermediate claim HxX: x ∈ X.
L179466
An exact proof term for the current goal is (HAsubX x HxA).
L179466
We prove the intermediate claim HnotI1: ¬ (apply_fun r11s x ∈ I1).
L179468
Assume Hr11sI1': apply_fun r11s x ∈ I1.
L179468
We prove the intermediate claim Hr11sI1I: apply_fun r11s x ∈ I1 ∩ I.
L179470
An exact proof term for the current goal is (binintersectI I1 I (apply_fun r11s x) Hr11sI1' Hr11sIx).
L179470
We prove the intermediate claim HxB12': x ∈ B12.
L179472
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r11s x0 ∈ I1 ∩ I) x HxA Hr11sI1I).
L179472
Apply FalseE to the current goal.
L179473
An exact proof term for the current goal is (HxnotB12 HxB12').
L179474
We prove the intermediate claim HnotI3: ¬ (apply_fun r11s x ∈ I3).
L179476
Assume Hr11sI3': apply_fun r11s x ∈ I3.
L179476
We prove the intermediate claim Hr11sI3I: apply_fun r11s x ∈ I3 ∩ I.
L179478
An exact proof term for the current goal is (binintersectI I3 I (apply_fun r11s x) Hr11sI3' Hr11sIx).
L179478
We prove the intermediate claim HxC12': x ∈ C12.
L179480
An exact proof term for the current goal is (SepI A (λx0 : set ⇒ apply_fun r11s x0 ∈ I3 ∩ I) x HxA Hr11sI3I).
L179480
Apply FalseE to the current goal.
L179481
An exact proof term for the current goal is (HxnotC12 HxC12').
L179482
We prove the intermediate claim Hr11sRx: apply_fun r11s x ∈ R.
L179484
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun r11s x) Hr11sIx).
L179484
We prove the intermediate claim Hm1R: minus_SNo 1 ∈ R.
L179486
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179486
We prove the intermediate claim H13R: one_third ∈ R.
L179488
An exact proof term for the current goal is one_third_in_R.
L179488
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L179490
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L179490
We prove the intermediate claim Hr11s_boundsI: Rle (minus_SNo 1) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) 1.
L179492
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r11s x) Hm1R real_1 Hr11sIx).
L179492
We prove the intermediate claim Hnlt_fx_m1: ¬ (Rlt (apply_fun r11s x) (minus_SNo 1)).
L179494
An exact proof term for the current goal is (RleE_nlt (minus_SNo 1) (apply_fun r11s x) (andEL (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_boundsI)).
L179495
We prove the intermediate claim Hnlt_1_fx: ¬ (Rlt 1 (apply_fun r11s x)).
L179497
An exact proof term for the current goal is (RleE_nlt (apply_fun r11s x) 1 (andER (Rle (minus_SNo 1) (apply_fun r11s x)) (Rle (apply_fun r11s x) 1) Hr11s_boundsI)).
L179498
We prove the intermediate claim HnotI1_cases: Rlt (apply_fun r11s x) (minus_SNo 1) ∨ Rlt (minus_SNo one_third) (apply_fun r11s x).
L179500
An exact proof term for the current goal is (closed_interval_not_mem_cases (minus_SNo 1) (minus_SNo one_third) (apply_fun r11s x) Hm1R Hm13R Hr11sRx HnotI1).
L179501
We prove the intermediate claim Hm13lt_fx: Rlt (minus_SNo one_third) (apply_fun r11s x).
L179503
Apply (HnotI1_cases (Rlt (minus_SNo one_third) (apply_fun r11s x))) to the current goal.
L179504
Assume Hbad: Rlt (apply_fun r11s x) (minus_SNo 1).
L179504
Apply FalseE to the current goal.
L179505
An exact proof term for the current goal is (Hnlt_fx_m1 Hbad).
L179507
Assume Hok: Rlt (minus_SNo one_third) (apply_fun r11s x).
L179507
An exact proof term for the current goal is Hok.
L179508
We prove the intermediate claim Hnot_fx_lt_m13: ¬ (Rlt (apply_fun r11s x) (minus_SNo one_third)).
L179510
An exact proof term for the current goal is (not_Rlt_sym (minus_SNo one_third) (apply_fun r11s x) Hm13lt_fx).
L179510
We prove the intermediate claim HnotI3_cases: Rlt (apply_fun r11s x) one_third ∨ Rlt 1 (apply_fun r11s x).
L179512
An exact proof term for the current goal is (closed_interval_not_mem_cases one_third 1 (apply_fun r11s x) H13R real_1 Hr11sRx HnotI3).
L179513
We prove the intermediate claim Hfx_lt_13: Rlt (apply_fun r11s x) one_third.
L179515
Apply (HnotI3_cases (Rlt (apply_fun r11s x) one_third)) to the current goal.
L179516
Assume Hok: Rlt (apply_fun r11s x) one_third.
L179516
An exact proof term for the current goal is Hok.
L179518
Assume Hbad: Rlt 1 (apply_fun r11s x).
L179518
Apply FalseE to the current goal.
L179519
An exact proof term for the current goal is (Hnlt_1_fx Hbad).
L179520
We prove the intermediate claim Hnot_13_lt_fx: ¬ (Rlt one_third (apply_fun r11s x)).
L179522
An exact proof term for the current goal is (not_Rlt_sym (apply_fun r11s x) one_third Hfx_lt_13).
L179522
We prove the intermediate claim Hr11sI0: apply_fun r11s x ∈ I0.
L179524
We prove the intermediate claim HI0_def: I0 = {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
Use reflexivity.
L179525
We prove the intermediate claim HxSep: apply_fun r11s x ∈ {t ∈ R|¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)}.
L179527
An exact proof term for the current goal is (SepI R (λt : set ⇒ ¬ (Rlt t (minus_SNo one_third)) ∧ ¬ (Rlt one_third t)) (apply_fun r11s x) Hr11sRx (andI (¬ (Rlt (apply_fun r11s x) (minus_SNo one_third))) (¬ (Rlt one_third (apply_fun r11s x))) Hnot_fx_lt_m13 Hnot_13_lt_fx)).
L179531
rewrite the current goal using HI0_def (from left to right).
L179532
An exact proof term for the current goal is HxSep.
L179533
We prove the intermediate claim Hu12funI0: function_on u12 X I0.
L179535
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 u12 Hu12contI0).
L179535
We prove the intermediate claim Hu12xI0: apply_fun u12 x ∈ I0.
L179537
An exact proof term for the current goal is (Hu12funI0 x HxX).
L179537
We prove the intermediate claim Hu12xR: apply_fun u12 x ∈ R.
L179539
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun u12 x) Hu12xI0).
L179539
We prove the intermediate claim Hm_u12x_R: minus_SNo (apply_fun u12 x) ∈ R.
L179541
An exact proof term for the current goal is (real_minus_SNo (apply_fun u12 x) Hu12xR).
L179541
rewrite the current goal using (Hr12_apply x HxA) (from left to right).
L179542
We prove the intermediate claim Hr12xR: add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) ∈ R.
L179544
An exact proof term for the current goal is (real_add_SNo (apply_fun r11s x) (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) (minus_SNo (apply_fun u12 x)) Hm_u12x_R).
L179546
We prove the intermediate claim H23R: two_thirds ∈ R.
L179548
An exact proof term for the current goal is two_thirds_in_R.
L179548
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L179550
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L179550
We prove the intermediate claim Hr11s_bounds0: Rle (minus_SNo one_third) (apply_fun r11s x) ∧ Rle (apply_fun r11s x) one_third.
L179552
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun r11s x) Hm13R H13R Hr11sI0).
L179552
We prove the intermediate claim Hu12_bounds0: Rle (minus_SNo one_third) (apply_fun u12 x) ∧ Rle (apply_fun u12 x) one_third.
L179554
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun u12 x) Hm13R H13R Hu12xI0).
L179554
We prove the intermediate claim Hr11s_le_13: Rle (apply_fun r11s x) one_third.
L179556
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun r11s x)) (Rle (apply_fun r11s x) one_third) Hr11s_bounds0).
L179559
We prove the intermediate claim Hr11s_ge_m13: Rle (minus_SNo one_third) (apply_fun r11s x).
L179561
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun r11s x)) (Rle (apply_fun r11s x) one_third) Hr11s_bounds0).
L179564
We prove the intermediate claim Hu12_le_13: Rle (apply_fun u12 x) one_third.
L179566
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun u12 x)) (Rle (apply_fun u12 x) one_third) Hu12_bounds0).
L179569
We prove the intermediate claim Hu12_ge_m13: Rle (minus_SNo one_third) (apply_fun u12 x).
L179571
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun u12 x)) (Rle (apply_fun u12 x) one_third) Hu12_bounds0).
L179574
We prove the intermediate claim Hm13_le_mu12: Rle (minus_SNo one_third) (minus_SNo (apply_fun u12 x)).
L179576
An exact proof term for the current goal is (Rle_minus_contra (apply_fun u12 x) one_third Hu12_le_13).
L179576
We prove the intermediate claim Hmu12_le_13: Rle (minus_SNo (apply_fun u12 x)) one_third.
L179578
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun u12 x)) (minus_SNo (minus_SNo one_third)).
L179579
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun u12 x) Hu12_ge_m13).
L179579
rewrite the current goal using (minus_SNo_minus_SNo_real one_third H13R) (from right to left) at position 1.
L179580
An exact proof term for the current goal is Htmp.
L179581
We prove the intermediate claim Hlow1: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))).
L179584
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun u12 x)) Hm13R Hm13R Hm_u12x_R Hm13_le_mu12).
L179585
We prove the intermediate claim Hlow2: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L179588
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) Hm13R (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) Hm_u12x_R Hr11s_ge_m13).
L179591
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L179594
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) Hlow1 Hlow2).
L179597
We prove the intermediate claim Hlow: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))).
L179599
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L179599
An exact proof term for the current goal is Hlow_tmp.
L179600
We prove the intermediate claim Hup1: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) one_third).
L179603
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r11s x) (minus_SNo (apply_fun u12 x)) one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) Hm_u12x_R H13R Hmu12_le_13).
L179605
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r11s x) one_third) (add_SNo one_third one_third).
L179608
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r11s x) one_third one_third (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun r11s x) Hr11sI0) H13R H13R Hr11s_le_13).
L179610
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo one_third one_third).
L179613
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) (add_SNo (apply_fun r11s x) one_third) (add_SNo one_third one_third) Hup1 Hup2).
L179616
We prove the intermediate claim Hdef23: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L179618
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) two_thirds.
L179620
rewrite the current goal using Hdef23 (from left to right) at position 1.
L179620
An exact proof term for the current goal is Hup_tmp.
L179621
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo two_thirds) two_thirds (add_SNo (apply_fun r11s x) (minus_SNo (apply_fun u12 x))) Hm23R H23R Hr12xR Hlow Hup).
L179624
Set r12s to be the term compose_fun A r12 (div_const_fun den).
L179625
We prove the intermediate claim Hr12s_cont: continuous_map A Ta I Ti r12s.
L179627
We prove the intermediate claim Hdivcont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
(*** structured: prove continuity into R, then restrict range to I ***)
L179629
An exact proof term for the current goal is (div_const_fun_continuous_pos den H23R H23pos).
L179629
We prove the intermediate claim Hr12s_contR: continuous_map A Ta R R_standard_topology r12s.
L179631
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology r12 (div_const_fun den) Hr12_cont Hdivcont).
L179632
We prove the intermediate claim Hr12s_I: ∀x : set, x ∈ A → apply_fun r12s x ∈ I.
L179634
Let x be given.
L179634
Assume HxA: x ∈ A.
L179634
We prove the intermediate claim Hr12xI2: apply_fun r12 x ∈ I2.
L179636
An exact proof term for the current goal is (Hr12_range x HxA).
L179636
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L179638
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L179638
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L179640
An exact proof term for the current goal is (real_minus_SNo den H23R).
L179640
We prove the intermediate claim Hr12xR: apply_fun r12 x ∈ R.
L179642
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun r12 x) Hr12xI2).
L179642
We prove the intermediate claim Hr12xS: SNo (apply_fun r12 x).
L179644
An exact proof term for the current goal is (real_SNo (apply_fun r12 x) Hr12xR).
L179644
We prove the intermediate claim Hbounds: Rle (minus_SNo den) (apply_fun r12 x) ∧ Rle (apply_fun r12 x) den.
L179646
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo den) den (apply_fun r12 x) HmdenR H23R Hr12xI2).
L179646
We prove the intermediate claim Hlo: Rle (minus_SNo den) (apply_fun r12 x).
L179648
An exact proof term for the current goal is (andEL (Rle (minus_SNo den) (apply_fun r12 x)) (Rle (apply_fun r12 x) den) Hbounds).
L179650
We prove the intermediate claim Hhi: Rle (apply_fun r12 x) den.
L179652
An exact proof term for the current goal is (andER (Rle (minus_SNo den) (apply_fun r12 x)) (Rle (apply_fun r12 x) den) Hbounds).
L179654
We prove the intermediate claim Hnlt_hi: ¬ (Rlt den (apply_fun r12 x)).
L179656
An exact proof term for the current goal is (RleE_nlt (apply_fun r12 x) den Hhi).
L179656
We prove the intermediate claim Hnlt_lo: ¬ (Rlt (apply_fun r12 x) (minus_SNo den)).
L179658
An exact proof term for the current goal is (RleE_nlt (minus_SNo den) (apply_fun r12 x) Hlo).
L179658
We prove the intermediate claim HyEq: apply_fun r12s x = div_SNo (apply_fun r12 x) den.
L179660
rewrite the current goal using (compose_fun_apply A r12 (div_const_fun den) x HxA) (from left to right).
L179660
rewrite the current goal using (div_const_fun_apply den (apply_fun r12 x) H23R Hr12xR) (from left to right).
Use reflexivity.
L179662
We prove the intermediate claim HyR: apply_fun r12s x ∈ R.
L179664
rewrite the current goal using HyEq (from left to right).
L179664
An exact proof term for the current goal is (real_div_SNo (apply_fun r12 x) Hr12xR den H23R).
L179665
We prove the intermediate claim HyS: SNo (apply_fun r12s x).
L179667
An exact proof term for the current goal is (real_SNo (apply_fun r12s x) HyR).
L179667
We prove the intermediate claim Hy_le_1: Rle (apply_fun r12s x) 1.
L179669
Apply (RleI (apply_fun r12s x) 1 HyR real_1) to the current goal.
L179669
We will prove ¬ (Rlt 1 (apply_fun r12s x)).
L179670
Assume H1lt: Rlt 1 (apply_fun r12s x).
L179671
We prove the intermediate claim H1lty: 1 < apply_fun r12s x.
L179673
An exact proof term for the current goal is (RltE_lt 1 (apply_fun r12s x) H1lt).
L179673
We prove the intermediate claim HmulLt: mul_SNo 1 den < mul_SNo (apply_fun r12s x) den.
L179675
An exact proof term for the current goal is (pos_mul_SNo_Lt' 1 (apply_fun r12s x) den SNo_1 HyS H23S H23pos H1lty).
L179675
We prove the intermediate claim HmulLt': den < mul_SNo (apply_fun r12s x) den.
L179677
rewrite the current goal using (mul_SNo_oneL den H23S) (from right to left) at position 1.
L179677
An exact proof term for the current goal is HmulLt.
L179678
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r12s x) den = apply_fun r12 x.
L179680
rewrite the current goal using HyEq (from left to right) at position 1.
L179680
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r12 x) den Hr12xS H23S H23ne0).
L179681
We prove the intermediate claim Hden_lt_r12x: den < apply_fun r12 x.
L179683
rewrite the current goal using HmulEq (from right to left).
L179683
An exact proof term for the current goal is HmulLt'.
L179684
We prove the intermediate claim Hbad: Rlt den (apply_fun r12 x).
L179686
An exact proof term for the current goal is (RltI den (apply_fun r12 x) H23R Hr12xR Hden_lt_r12x).
L179686
An exact proof term for the current goal is (Hnlt_hi Hbad).
L179687
We prove the intermediate claim Hm1_le_y: Rle (minus_SNo 1) (apply_fun r12s x).
L179689
Apply (RleI (minus_SNo 1) (apply_fun r12s x) Hm1R HyR) to the current goal.
L179689
We will prove ¬ (Rlt (apply_fun r12s x) (minus_SNo 1)).
L179690
Assume Hylt: Rlt (apply_fun r12s x) (minus_SNo 1).
L179691
We prove the intermediate claim Hylts: apply_fun r12s x < minus_SNo 1.
L179693
An exact proof term for the current goal is (RltE_lt (apply_fun r12s x) (minus_SNo 1) Hylt).
L179693
We prove the intermediate claim HmulLt: mul_SNo (apply_fun r12s x) den < mul_SNo (minus_SNo 1) den.
L179695
An exact proof term for the current goal is (pos_mul_SNo_Lt' (apply_fun r12s x) (minus_SNo 1) den HyS (SNo_minus_SNo 1 SNo_1) H23S H23pos Hylts).
L179696
We prove the intermediate claim HmulEq: mul_SNo (apply_fun r12s x) den = apply_fun r12 x.
L179698
rewrite the current goal using HyEq (from left to right) at position 1.
L179698
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun r12 x) den Hr12xS H23S H23ne0).
L179699
We prove the intermediate claim HrhsEq: mul_SNo (minus_SNo 1) den = minus_SNo den.
L179701
rewrite the current goal using (mul_SNo_minus_distrL 1 den SNo_1 H23S) (from left to right) at position 1.
L179701
rewrite the current goal using (mul_SNo_oneL den H23S) (from left to right) at position 1.
Use reflexivity.
L179703
We prove the intermediate claim Hr12x_lt_mden: apply_fun r12 x < minus_SNo den.
L179705
rewrite the current goal using HmulEq (from right to left).
L179705
rewrite the current goal using HrhsEq (from right to left).
L179706
An exact proof term for the current goal is HmulLt.
L179707
We prove the intermediate claim Hbad: Rlt (apply_fun r12 x) (minus_SNo den).
L179709
An exact proof term for the current goal is (RltI (apply_fun r12 x) (minus_SNo den) Hr12xR HmdenR Hr12x_lt_mden).
L179709
An exact proof term for the current goal is (Hnlt_lo Hbad).
L179710
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (apply_fun r12s x) Hm1R real_1 HyR Hm1_le_y Hy_le_1).
L179712
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology r12s I Hr12s_contR HIcR Hr12s_I).
L179714
We prove the intermediate claim Hexfn: ∃fn : set, function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) ∧ uniform_cauchy_metric X R R_bounded_metric fn.
(*** pending: continue the iteration further and prove uniform convergence ***)
L179726
Set I0 to be the term closed_interval (minus_SNo one_third) one_third.
(*** TeX Step II (in progress): define a full correction sequence by nat_primrec and prove it has the needed properties. ***)
L179728
Set T0 to be the term closed_interval_topology (minus_SNo one_third) one_third.
(*** We keep the already-built finite scaffold above, but the infinite sequence is defined independently. ***)
L179729
Set u_of to be the term (λr : set ⇒ Eps_i (λu : set ⇒ continuous_map X Tx I0 T0 u ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun u x = one_third))).
L179736
We prove the intermediate claim Hu_of_prop: ∀r : set, continuous_map A Ta I Ti r → continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
(*** helper: Step I witness extracted via Eps_i ***)
L179744
Let r be given.
L179744
Assume Hr: continuous_map A Ta I Ti r.
L179744
Set P to be the term (λu : set ⇒ continuous_map X Tx I0 T0 u ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun u x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun u x = one_third)).
L179750
We prove the intermediate claim Hexu: ∃u : set, P u.
L179752
An exact proof term for the current goal is (Tietze_stepI_g_exists X Tx A r Hnorm HA Hr).
L179752
Apply Hexu to the current goal.
L179753
Let u be given.
L179754
Assume Hu.
L179754
We prove the intermediate claim HP: P (Eps_i P).
L179756
An exact proof term for the current goal is (Eps_i_ax P u Hu).
L179756
We prove the intermediate claim Hu_eq: u_of r = Eps_i P.
Use reflexivity.
L179758
rewrite the current goal using Hu_eq (from left to right).
L179759
An exact proof term for the current goal is HP.
L179760
Set g0g to be the term graph X (λx : set ⇒ apply_fun g0 x).
L179762
Set BaseState to be the term (g0g,(f1s,den)).
(*** base stage: graphify g0 so that it lies in function_space X R ***)
L179763
Set StepState to be the term (λn st : set ⇒ (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun ((st 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo ((st 1) 1) den))).
L179779
Set State to be the term (λn : set ⇒ nat_primrec BaseState StepState n).
L179780
Set fn to be the term graph ω (λn : set ⇒ (State n) 0).
L179781
We use fn to witness the existential quantifier.
L179782
We prove the intermediate claim HInv_cpos: ∀n : set, n ∈ ω → ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
(*** helper: scalar component stays positive ***)
L179785
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L179786
Apply nat_ind to the current goal.
L179787
We will prove ((State 0) 1) 1 ∈ R ∧ 0 < ((State 0) 1) 1.
L179787
We prove the intermediate claim H0: State 0 = BaseState.
L179789
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L179789
rewrite the current goal using H0 (from left to right).
L179790
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L179792
rewrite the current goal using HBase (from left to right).
L179793
We prove the intermediate claim HtEq: ((g0g,(f1s,den)) 1) 1 = den.
L179795
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L179795
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L179797
Apply andI to the current goal.
L179799
rewrite the current goal using HtEq (from left to right).
L179799
An exact proof term for the current goal is HdenR.
L179801
rewrite the current goal using HtEq (from left to right).
L179801
An exact proof term for the current goal is HdenPos.
L179803
Let n be given.
L179803
Assume HnNat: nat_p n.
L179803
Assume IH: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L179804
We will prove ((State (ordsucc n)) 1) 1 ∈ R ∧ 0 < ((State (ordsucc n)) 1) 1.
L179805
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L179807
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L179807
rewrite the current goal using HS (from left to right).
L179808
Set st to be the term State n.
L179809
Set g to be the term st 0.
L179810
Set rpack to be the term st 1.
L179811
Set r to be the term rpack 0.
L179812
Set c to be the term rpack 1.
L179813
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L179816
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L179821
Set cNew to be the term mul_SNo c den.
L179822
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L179824
We prove the intermediate claim HtEq: ((StepState n st) 1) 1 = cNew.
L179826
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L179827
rewrite the current goal using Hdef (from left to right).
L179827
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L179828
rewrite the current goal using Hinner (from left to right).
L179829
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L179830
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L179832
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L179834
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) IH).
L179834
We prove the intermediate claim Hcpos0: 0 < ((State n) 1) 1.
L179836
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) IH).
L179836
We prove the intermediate claim HcR: c ∈ R.
L179838
rewrite the current goal using Hc_eq (from left to right).
L179838
An exact proof term for the current goal is HcR0.
L179838
We prove the intermediate claim Hcpos: 0 < c.
L179840
rewrite the current goal using Hc_eq (from left to right).
L179840
An exact proof term for the current goal is Hcpos0.
L179840
We prove the intermediate claim HcS: SNo c.
L179842
An exact proof term for the current goal is (real_SNo c HcR).
L179842
We prove the intermediate claim HdenS: SNo den.
L179844
An exact proof term for the current goal is (real_SNo den HdenR).
L179844
We prove the intermediate claim HcNewR: cNew ∈ R.
L179846
An exact proof term for the current goal is (real_mul_SNo c HcR den HdenR).
L179846
We prove the intermediate claim HcNewPos: 0 < cNew.
L179848
An exact proof term for the current goal is (mul_SNo_pos_pos c den HcS HdenS Hcpos HdenPos).
L179848
Apply andI to the current goal.
L179850
rewrite the current goal using HtEq (from left to right).
L179850
An exact proof term for the current goal is HcNewR.
L179852
rewrite the current goal using HtEq (from left to right).
L179852
An exact proof term for the current goal is HcNewPos.
L179853
Let n be given.
L179854
Assume HnO: n ∈ ω.
L179854
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L179855
We prove the intermediate claim HInv_c_lt1: ∀n : set, n ∈ ω → ((State n) 1) 1 < 1.
(*** helper: scalar component is always strictly below 1 ***)
L179858
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ((State n) 1) 1 < 1.
L179859
Apply nat_ind to the current goal.
L179860
We will prove ((State 0) 1) 1 < 1.
L179860
We prove the intermediate claim H0: State 0 = BaseState.
L179862
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L179862
rewrite the current goal using H0 (from left to right).
L179863
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L179865
rewrite the current goal using HBase (from left to right).
L179866
We prove the intermediate claim HtEq: ((g0g,(f1s,den)) 1) 1 = den.
L179868
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L179868
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L179870
rewrite the current goal using HtEq (from left to right).
L179871
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L179873
rewrite the current goal using HdenDef (from left to right).
L179874
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L179876
Let n be given.
L179876
Assume HnNat: nat_p n.
L179876
Assume IH: ((State n) 1) 1 < 1.
L179877
We will prove ((State (ordsucc n)) 1) 1 < 1.
L179878
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L179880
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L179880
rewrite the current goal using HS (from left to right).
L179881
Set st to be the term State n.
L179882
Set g to be the term st 0.
L179883
Set rpack to be the term st 1.
L179884
Set r to be the term rpack 0.
L179885
Set c to be the term rpack 1.
L179886
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L179889
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L179894
Set cNew to be the term mul_SNo c den.
L179895
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L179897
We prove the intermediate claim HtEq: ((StepState n st) 1) 1 = cNew.
L179899
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L179900
rewrite the current goal using Hdef (from left to right).
L179900
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L179901
rewrite the current goal using Hinner (from left to right).
L179902
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L179903
rewrite the current goal using HtEq (from left to right).
L179904
We prove the intermediate claim HnO: n ∈ ω.
L179906
An exact proof term for the current goal is (nat_p_omega n HnNat).
L179906
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L179908
An exact proof term for the current goal is (HInv_cpos n HnO).
L179908
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L179910
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L179910
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L179912
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L179912
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L179914
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L179914
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L179916
We prove the intermediate claim HdenLt1: den < 1.
L179918
rewrite the current goal using HdenDef (from left to right).
L179918
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L179919
We prove the intermediate claim HdenS: SNo den.
L179921
An exact proof term for the current goal is (real_SNo den HdenR).
L179921
We prove the intermediate claim HmulLt: mul_SNo den (((State n) 1) 1) < ((State n) 1) 1.
L179923
An exact proof term for the current goal is (mul_SNo_Lt1_pos_Lt den (((State n) 1) 1) HdenS HcS HdenLt1 Hcpos).
L179923
We prove the intermediate claim HmulEq: mul_SNo den (((State n) 1) 1) = mul_SNo (((State n) 1) 1) den.
L179925
An exact proof term for the current goal is (mul_SNo_com den (((State n) 1) 1) HdenS HcS).
L179925
We prove the intermediate claim HcNewLt: cNew < ((State n) 1) 1.
L179927
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L179928
rewrite the current goal using HcEq (from left to right) at position 1.
L179929
rewrite the current goal using HmulEq (from right to left) at position 1.
L179930
An exact proof term for the current goal is HmulLt.
L179931
We prove the intermediate claim HcNewR: cNew ∈ R.
L179933
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L179934
rewrite the current goal using HcEq (from left to right).
L179935
An exact proof term for the current goal is (real_mul_SNo (((State n) 1) 1) HcR den HdenR).
L179936
We prove the intermediate claim HcNewS: SNo cNew.
L179938
An exact proof term for the current goal is (real_SNo cNew HcNewR).
L179938
An exact proof term for the current goal is (SNoLt_tra cNew (((State n) 1) 1) 1 HcNewS HcS SNo_1 HcNewLt IH).
L179944
Let n be given.
L179945
Assume HnO: n ∈ ω.
L179945
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L179946
We prove the intermediate claim HInv_r_contI: ∀n : set, n ∈ ω → continuous_map A Ta I Ti (((State n) 1) 0).
(*** helper: residual component stays a continuous I-valued map on A ***)
L179949
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → continuous_map A Ta I Ti (((State n) 1) 0).
L179950
Apply nat_ind to the current goal.
L179951
We will prove continuous_map A Ta I Ti (((State 0) 1) 0).
L179951
We prove the intermediate claim H0: State 0 = BaseState.
L179953
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L179953
rewrite the current goal using H0 (from left to right).
L179954
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L179956
rewrite the current goal using HBase (from left to right).
L179957
We prove the intermediate claim Hr0eq: (((g0g,(f1s,den)) 1) 0) = f1s.
L179959
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L179959
rewrite the current goal using (tuple_2_0_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L179961
rewrite the current goal using Hr0eq (from left to right).
L179962
An exact proof term for the current goal is Hf1s_cont.
L179964
Let n be given.
L179964
Assume HnNat: nat_p n.
L179964
Assume IH: continuous_map A Ta I Ti (((State n) 1) 0).
L179965
We will prove continuous_map A Ta I Ti (((State (ordsucc n)) 1) 0).
L179966
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L179968
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L179968
rewrite the current goal using HS (from left to right).
L179969
Set st to be the term State n.
L179970
Set g to be the term st 0.
L179971
Set rpack to be the term st 1.
L179972
Set r to be the term rpack 0.
L179973
Set c to be the term rpack 1.
L179974
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L179977
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L179982
Set cNew to be the term mul_SNo c den.
L179983
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L179985
We prove the intermediate claim HrEq: ((StepState n st) 1) 0 = rNew.
L179987
We prove the intermediate claim Hinner: (StepState n st) 1 = (rNew,cNew).
L179988
rewrite the current goal using Hdef (from left to right).
L179988
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L179989
rewrite the current goal using Hinner (from left to right).
L179990
An exact proof term for the current goal is (tuple_2_0_eq rNew cNew).
L179991
rewrite the current goal using HrEq (from left to right).
L179992
We prove the intermediate claim HIcR: I ⊆ R.
(*** continuity of rNew into I, via range restriction ***)
L179995
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L179995
We prove the intermediate claim HTiEq: Ti = subspace_topology R R_standard_topology I.
Use reflexivity.
L179997
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L179999
We prove the intermediate claim HrEq0: r = ((State n) 1) 0.
Use reflexivity.
L180000
rewrite the current goal using HrEq0 (from left to right).
L180001
An exact proof term for the current goal is IH.
L180002
We prove the intermediate claim Hr_contR: continuous_map A Ta R R_standard_topology r.
L180004
An exact proof term for the current goal is (continuous_map_range_expand A Ta I Ti R R_standard_topology r Hr_contI HIcR R_standard_topology_is_topology_local HTiEq).
L180005
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L180012
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L180012
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L180017
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L180023
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L180025
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L180029
We prove the intermediate claim HI0cR: I0 ⊆ R.
L180031
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L180031
We prove the intermediate claim HT0eq: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L180033
We prove the intermediate claim Hu_contR: continuous_map X Tx R R_standard_topology (u_of r).
L180035
An exact proof term for the current goal is (continuous_map_range_expand X Tx I0 T0 R R_standard_topology (u_of r) Hu_contI0 HI0cR R_standard_topology_is_topology_local HT0eq).
L180036
We prove the intermediate claim HTx: topology_on X Tx.
L180038
An exact proof term for the current goal is (normal_space_topology_on X Tx Hnorm).
L180038
We prove the intermediate claim HAsubX: A ⊆ X.
L180040
An exact proof term for the current goal is (closed_in_subset X Tx A HA).
L180040
We prove the intermediate claim HuA_contR: continuous_map A Ta R R_standard_topology (u_of r).
L180042
An exact proof term for the current goal is (continuous_on_subspace X Tx R R_standard_topology (u_of r) A HTx HAsubX Hu_contR).
L180042
We prove the intermediate claim Hneg_cont: continuous_map R R_standard_topology R R_standard_topology neg_fun.
L180044
An exact proof term for the current goal is neg_fun_continuous.
L180044
We prove the intermediate claim HnegA_contR: continuous_map A Ta R R_standard_topology (compose_fun A (u_of r) neg_fun).
L180046
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology (u_of r) neg_fun HuA_contR Hneg_cont).
L180047
We prove the intermediate claim Hpair_cont: continuous_map A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) (pair_map A r (compose_fun A (u_of r) neg_fun)).
L180051
An exact proof term for the current goal is (maps_into_products_axiom A Ta R R_standard_topology R R_standard_topology r (compose_fun A (u_of r) neg_fun) Hr_contR HnegA_contR).
L180052
We prove the intermediate claim Hadd_cont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L180056
An exact proof term for the current goal is add_fun_R_continuous.
L180056
We prove the intermediate claim Hh_cont: continuous_map A Ta R R_standard_topology (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R).
L180059
An exact proof term for the current goal is (composition_continuous A Ta (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R Hpair_cont Hadd_cont).
L180063
We prove the intermediate claim Hdiv_cont: continuous_map R R_standard_topology R R_standard_topology (div_const_fun den).
L180065
An exact proof term for the current goal is (div_const_fun_continuous_pos den HdenR HdenPos).
L180065
We prove the intermediate claim HrNew_contR: continuous_map A Ta R R_standard_topology rNew.
L180067
An exact proof term for the current goal is (composition_continuous A Ta R R_standard_topology R R_standard_topology (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den) Hh_cont Hdiv_cont).
L180070
We prove the intermediate claim HrNew_img: ∀x : set, x ∈ A → apply_fun rNew x ∈ I.
L180072
Let x be given.
L180072
Assume HxA: x ∈ A.
L180072
Set rNum to be the term compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R.
L180073
Set I2 to be the term closed_interval (minus_SNo den) den.
L180074
We prove the intermediate claim HrNumI2: apply_fun rNum x ∈ I2.
L180076
We prove the intermediate claim HxX: x ∈ X.
(*** proof scaffold: rNum x = r x - u_of r x, then show bounds in [-den,den] ***)
L180078
An exact proof term for the current goal is (HAsubX x HxA).
L180078
We prove the intermediate claim Hfun_r: function_on r A I.
L180080
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r Hr_contI).
L180080
We prove the intermediate claim HrxI: apply_fun r x ∈ I.
L180082
An exact proof term for the current goal is (Hfun_r x HxA).
L180082
We prove the intermediate claim HrxR: apply_fun r x ∈ R.
L180084
An exact proof term for the current goal is (HIcR (apply_fun r x) HrxI).
L180084
We prove the intermediate claim Hfun_u: function_on (u_of r) X I0.
L180086
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L180086
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L180088
An exact proof term for the current goal is (Hfun_u x HxX).
L180088
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L180090
An exact proof term for the current goal is (HI0cR (apply_fun (u_of r) x) HuxI0).
L180090
We prove the intermediate claim HrNumEq: apply_fun rNum x = add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)).
L180093
An exact proof term for the current goal is (add_of_pair_map_neg_apply A r (u_of r) x HxA HrxR HuxR).
L180093
Apply (xm (Rlt (apply_fun r x) (minus_SNo one_third))) to the current goal.
L180095
Assume Hr_lt_left: Rlt (apply_fun r x) (minus_SNo one_third).
L180095
We prove the intermediate claim Hu_left_eq: ∀y : set, y ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y = minus_SNo one_third.
L180099
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 (u_of r)) (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y0 = minus_SNo one_third) Hu_left).
L180102
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L180104
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L180104
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L180106
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L180106
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L180108
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) Hm1R real_1 HrxI).
L180108
We prove the intermediate claim Hlo: Rle (minus_SNo 1) (apply_fun r x).
L180110
An exact proof term for the current goal is (andEL (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L180110
We prove the intermediate claim Hhi: Rle (apply_fun r x) (minus_SNo one_third).
L180112
An exact proof term for the current goal is (Rlt_implies_Rle (apply_fun r x) (minus_SNo one_third) Hr_lt_left).
L180112
We prove the intermediate claim HrxIleft: apply_fun r x ∈ closed_interval (minus_SNo 1) (minus_SNo one_third).
L180114
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) (minus_SNo one_third) (apply_fun r x) Hm1R Hm13R HrxR Hlo Hhi).
L180115
We prove the intermediate claim HrxV: apply_fun r x ∈ (closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I.
L180117
An exact proof term for the current goal is (binintersectI (closed_interval (minus_SNo 1) (minus_SNo one_third)) I (apply_fun r x) HrxIleft HrxI).
L180117
We prove the intermediate claim HxPre: x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I).
L180119
An exact proof term for the current goal is (SepI A (λz : set ⇒ apply_fun r z ∈ ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I)) x HxA HrxV).
L180120
We prove the intermediate claim HuxEq: apply_fun (u_of r) x = minus_SNo one_third.
L180122
An exact proof term for the current goal is (Hu_left_eq x HxPre).
L180122
We prove the intermediate claim HrNumEqL: apply_fun rNum x = add_SNo (apply_fun r x) one_third.
L180124
rewrite the current goal using HrNumEq (from left to right).
L180124
rewrite the current goal using HuxEq (from left to right).
L180125
rewrite the current goal using (minus_SNo_minus_SNo_R one_third one_third_in_R) (from left to right) at position 1.
Use reflexivity.
L180127
rewrite the current goal using HrNumEqL (from left to right).
L180128
We prove the intermediate claim H13R: one_third ∈ R.
L180130
An exact proof term for the current goal is one_third_in_R.
L180130
We prove the intermediate claim H13S: SNo one_third.
L180132
An exact proof term for the current goal is (real_SNo one_third H13R).
L180132
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L180134
We prove the intermediate claim HmdenR: (minus_SNo den) ∈ R.
L180136
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L180136
We prove the intermediate claim HsumR: add_SNo (apply_fun r x) one_third ∈ R.
L180138
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR one_third H13R).
L180138
We prove the intermediate claim Hlow_tmp: Rle (add_SNo (minus_SNo 1) one_third) (add_SNo (apply_fun r x) one_third).
L180140
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo 1) (apply_fun r x) one_third Hm1R HrxR H13R Hlo).
L180140
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) one_third).
L180142
rewrite the current goal using HdenDef (from left to right) at position 1.
L180142
rewrite the current goal using minus_two_thirds_eq_minus1_plus_one_third (from left to right) at position 1.
L180143
An exact proof term for the current goal is Hlow_tmp.
L180144
We prove the intermediate claim Hup0_tmp: Rle (add_SNo (apply_fun r x) one_third) (add_SNo (minus_SNo one_third) one_third).
L180146
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) (minus_SNo one_third) one_third HrxR Hm13R H13R Hhi).
L180146
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r x) one_third) 0.
L180148
We prove the intermediate claim HtR: add_SNo (minus_SNo one_third) one_third ∈ R.
L180149
An exact proof term for the current goal is (real_add_SNo (minus_SNo one_third) Hm13R one_third H13R).
L180149
We prove the intermediate claim HtEq0: add_SNo (minus_SNo one_third) one_third = 0.
L180151
An exact proof term for the current goal is (add_SNo_minus_SNo_linv one_third H13S).
L180151
We prove the intermediate claim Ht0: Rle (add_SNo (minus_SNo one_third) one_third) 0.
L180153
rewrite the current goal using HtEq0 (from left to right).
L180153
An exact proof term for the current goal is (Rle_refl 0 real_0).
L180154
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) one_third) (add_SNo (minus_SNo one_third) one_third) 0 Hup0_tmp Ht0).
L180157
We prove the intermediate claim H0leDen: Rle 0 den.
L180159
rewrite the current goal using HdenDef (from left to right) at position 1.
L180159
An exact proof term for the current goal is Rle_0_two_thirds.
L180160
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) one_third) den.
L180162
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) one_third) 0 den Hup0 H0leDen).
L180162
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) one_third) HmdenR HdenR HsumR Hlow Hup).
L180165
Assume Hr_not_lt_left: ¬ (Rlt (apply_fun r x) (minus_SNo one_third)).
L180165
Apply (xm (Rlt one_third (apply_fun r x))) to the current goal.
L180167
Assume Hr_lt_right: Rlt one_third (apply_fun r x).
L180167
We prove the intermediate claim Hu_right_eq: ∀y : set, y ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) y = one_third.
L180171
An exact proof term for the current goal is (andER (continuous_map X Tx I0 T0 (u_of r) ∧ (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) y0 = minus_SNo one_third)) (∀y0 : set, y0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) y0 = one_third) Hu_pack).
L180177
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L180179
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) (real_minus_SNo 1 real_1) real_1 HrxI).
L180179
We prove the intermediate claim Hhi: Rle (apply_fun r x) 1.
L180181
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L180181
We prove the intermediate claim Hlo: Rle one_third (apply_fun r x).
L180183
An exact proof term for the current goal is (Rlt_implies_Rle one_third (apply_fun r x) Hr_lt_right).
L180183
We prove the intermediate claim HrxIright: apply_fun r x ∈ closed_interval one_third 1.
L180185
An exact proof term for the current goal is (closed_intervalI_of_Rle one_third 1 (apply_fun r x) one_third_in_R real_1 HrxR Hlo Hhi).
L180186
We prove the intermediate claim HrxV: apply_fun r x ∈ (closed_interval one_third 1) ∩ I.
L180188
An exact proof term for the current goal is (binintersectI (closed_interval one_third 1) I (apply_fun r x) HrxIright HrxI).
L180188
We prove the intermediate claim HxPre: x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I).
L180190
An exact proof term for the current goal is (SepI A (λz : set ⇒ apply_fun r z ∈ ((closed_interval one_third 1) ∩ I)) x HxA HrxV).
L180191
We prove the intermediate claim HuxEq: apply_fun (u_of r) x = one_third.
L180193
An exact proof term for the current goal is (Hu_right_eq x HxPre).
L180193
We prove the intermediate claim HrNumEqR: apply_fun rNum x = add_SNo (apply_fun r x) (minus_SNo one_third).
L180195
rewrite the current goal using HrNumEq (from left to right).
L180195
rewrite the current goal using HuxEq (from left to right).
Use reflexivity.
L180197
rewrite the current goal using HrNumEqR (from left to right).
L180198
We prove the intermediate claim H13R: one_third ∈ R.
L180200
An exact proof term for the current goal is one_third_in_R.
L180200
We prove the intermediate claim H23R: two_thirds ∈ R.
L180202
An exact proof term for the current goal is two_thirds_in_R.
L180202
We prove the intermediate claim Hm23R: minus_SNo two_thirds ∈ R.
L180204
An exact proof term for the current goal is (real_minus_SNo two_thirds H23R).
L180204
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L180206
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L180206
We prove the intermediate claim HrxB: Rle (minus_SNo 1) (apply_fun r x) ∧ Rle (apply_fun r x) 1.
L180208
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo 1) 1 (apply_fun r x) (real_minus_SNo 1 real_1) real_1 HrxI).
L180208
We prove the intermediate claim Hrxle1: Rle (apply_fun r x) 1.
L180210
An exact proof term for the current goal is (andER (Rle (minus_SNo 1) (apply_fun r x)) (Rle (apply_fun r x) 1) HrxB).
L180210
We prove the intermediate claim HrNumR: add_SNo (apply_fun r x) (minus_SNo one_third) ∈ R.
L180212
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR (minus_SNo one_third) Hm13R).
L180212
We prove the intermediate claim H0le_rNum_tmp: Rle (add_SNo one_third (minus_SNo one_third)) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L180215
An exact proof term for the current goal is (Rle_add_SNo_1 one_third (apply_fun r x) (minus_SNo one_third) H13R HrxR Hm13R Hlo).
L180215
We prove the intermediate claim H13S: SNo one_third.
L180217
An exact proof term for the current goal is (real_SNo one_third H13R).
L180217
We prove the intermediate claim H0le_rNum: Rle 0 (add_SNo (apply_fun r x) (minus_SNo one_third)).
L180219
rewrite the current goal using (add_SNo_minus_SNo_rinv one_third H13S) (from right to left) at position 1.
L180219
An exact proof term for the current goal is H0le_rNum_tmp.
L180220
We prove the intermediate claim Hm23le0: Rle (minus_SNo two_thirds) 0.
L180222
An exact proof term for the current goal is Rle_minus_two_thirds_0.
L180222
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L180224
We prove the intermediate claim Hlow0: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L180226
An exact proof term for the current goal is (Rle_tra (minus_SNo two_thirds) 0 (add_SNo (apply_fun r x) (minus_SNo one_third)) Hm23le0 H0le_rNum).
L180226
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) (minus_SNo one_third)).
L180228
rewrite the current goal using HdenDef (from left to right) at position 1.
L180228
An exact proof term for the current goal is Hlow0.
L180229
We prove the intermediate claim Hup_tmp: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)).
L180232
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) 1 (minus_SNo one_third) HrxR real_1 Hm13R Hrxle1).
L180232
We prove the intermediate claim Hup1: Rle (add_SNo 1 (minus_SNo one_third)) two_thirds.
L180234
rewrite the current goal using two_thirds_eq_1_minus_one_third (from right to left) at position 1.
L180234
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L180235
We prove the intermediate claim Hup0: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) two_thirds.
L180237
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo one_third)) (add_SNo 1 (minus_SNo one_third)) two_thirds Hup_tmp Hup1).
L180240
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) (minus_SNo one_third)) den.
L180242
rewrite the current goal using HdenDef (from left to right) at position 1.
L180242
An exact proof term for the current goal is Hup0.
L180243
We prove the intermediate claim HmdenR: minus_SNo den ∈ R.
L180245
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L180245
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) (minus_SNo one_third)) HmdenR HdenR HrNumR Hlow Hup).
L180249
Assume Hr_not_lt_right: ¬ (Rlt one_third (apply_fun r x)).
L180249
We prove the intermediate claim Hm13R: (minus_SNo one_third) ∈ R.
L180251
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L180251
We prove the intermediate claim Hm13_le_rx: Rle (minus_SNo one_third) (apply_fun r x).
L180253
Apply (RleI (minus_SNo one_third) (apply_fun r x) Hm13R HrxR) to the current goal.
L180253
We will prove ¬ (Rlt (apply_fun r x) (minus_SNo one_third)).
L180254
An exact proof term for the current goal is Hr_not_lt_left.
L180255
We prove the intermediate claim Hrx_le_13: Rle (apply_fun r x) one_third.
L180257
Apply (RleI (apply_fun r x) one_third HrxR one_third_in_R) to the current goal.
L180257
We will prove ¬ (Rlt one_third (apply_fun r x)).
L180258
An exact proof term for the current goal is Hr_not_lt_right.
L180259
We prove the intermediate claim HrxI0: apply_fun r x ∈ closed_interval (minus_SNo one_third) one_third.
L180261
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo one_third) one_third (apply_fun r x) Hm13R one_third_in_R HrxR Hm13_le_rx Hrx_le_13).
L180262
rewrite the current goal using HrNumEq (from left to right).
L180263
We prove the intermediate claim H13R: one_third ∈ R.
L180265
An exact proof term for the current goal is one_third_in_R.
L180265
We prove the intermediate claim H23R: two_thirds ∈ R.
L180267
An exact proof term for the current goal is two_thirds_in_R.
L180267
We prove the intermediate claim Hux_bounds: Rle (minus_SNo one_third) (apply_fun (u_of r) x) ∧ Rle (apply_fun (u_of r) x) one_third.
L180269
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun (u_of r) x) Hm13R one_third_in_R HuxI0).
L180270
We prove the intermediate claim Hm13_le_ux: Rle (minus_SNo one_third) (apply_fun (u_of r) x).
L180272
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hux_bounds).
L180273
We prove the intermediate claim Hux_le_13: Rle (apply_fun (u_of r) x) one_third.
L180275
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hux_bounds).
L180276
We prove the intermediate claim Hm13_le_negux: Rle (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x)).
L180278
An exact proof term for the current goal is (Rle_minus_contra (apply_fun (u_of r) x) one_third Hux_le_13).
L180278
We prove the intermediate claim Hnegux_le_13: Rle (minus_SNo (apply_fun (u_of r) x)) one_third.
L180280
We prove the intermediate claim Htmp: Rle (minus_SNo (apply_fun (u_of r) x)) (minus_SNo (minus_SNo one_third)).
L180281
An exact proof term for the current goal is (Rle_minus_contra (minus_SNo one_third) (apply_fun (u_of r) x) Hm13_le_ux).
L180281
We prove the intermediate claim Hmid: Rle (minus_SNo (minus_SNo one_third)) one_third.
L180283
rewrite the current goal using (minus_SNo_minus_SNo_R one_third one_third_in_R) (from left to right) at position 1.
L180283
An exact proof term for the current goal is (Rle_refl one_third one_third_in_R).
L180284
An exact proof term for the current goal is (Rle_tra (minus_SNo (apply_fun (u_of r) x)) (minus_SNo (minus_SNo one_third)) one_third Htmp Hmid).
L180288
We prove the intermediate claim Hm13R2: minus_SNo one_third ∈ R.
L180290
An exact proof term for the current goal is Hm13R.
L180290
We prove the intermediate claim HneguxR: minus_SNo (apply_fun (u_of r) x) ∈ R.
L180292
An exact proof term for the current goal is (real_minus_SNo (apply_fun (u_of r) x) HuxR).
L180292
We prove the intermediate claim HrNumR: add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) ∈ R.
L180294
An exact proof term for the current goal is (real_add_SNo (apply_fun r x) HrxR (minus_SNo (apply_fun (u_of r) x)) HneguxR).
L180294
We prove the intermediate claim Hlow0a: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))).
L180297
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo one_third) (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x)) Hm13R2 Hm13R2 HneguxR Hm13_le_negux).
L180298
We prove the intermediate claim Hlow0b: Rle (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L180301
An exact proof term for the current goal is (Rle_add_SNo_1 (minus_SNo one_third) (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) Hm13R2 HrxR HneguxR Hm13_le_rx).
L180302
We prove the intermediate claim Hlow0: Rle (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L180305
An exact proof term for the current goal is (Rle_tra (add_SNo (minus_SNo one_third) (minus_SNo one_third)) (add_SNo (minus_SNo one_third) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) Hlow0a Hlow0b).
L180308
We prove the intermediate claim Hlow2: Rle (minus_SNo two_thirds) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L180311
rewrite the current goal using minus_two_thirds_eq (from left to right) at position 1.
L180311
An exact proof term for the current goal is Hlow0.
L180312
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L180314
We prove the intermediate claim Hlow: Rle (minus_SNo den) (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))).
L180317
rewrite the current goal using HdenDef (from left to right) at position 1.
L180317
An exact proof term for the current goal is Hlow2.
L180318
We prove the intermediate claim Hup0a: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) one_third).
L180321
An exact proof term for the current goal is (Rle_add_SNo_2 (apply_fun r x) (minus_SNo (apply_fun (u_of r) x)) one_third HrxR HneguxR H13R Hnegux_le_13).
L180322
We prove the intermediate claim Hup0b: Rle (add_SNo (apply_fun r x) one_third) (add_SNo one_third one_third).
L180324
An exact proof term for the current goal is (Rle_add_SNo_1 (apply_fun r x) one_third one_third HrxR H13R H13R Hrx_le_13).
L180324
We prove the intermediate claim Hup0c: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo one_third one_third).
L180326
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo (apply_fun r x) one_third) (add_SNo one_third one_third) Hup0a Hup0b).
L180329
We prove the intermediate claim Htwo_def: two_thirds = add_SNo one_third one_third.
Use reflexivity.
L180331
We prove the intermediate claim Hup1: Rle (add_SNo one_third one_third) two_thirds.
L180333
rewrite the current goal using Htwo_def (from right to left) at position 1.
L180333
An exact proof term for the current goal is (Rle_refl two_thirds H23R).
L180334
We prove the intermediate claim Hup2: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) two_thirds.
L180336
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) (add_SNo one_third one_third) two_thirds Hup0c Hup1).
L180339
We prove the intermediate claim Hup: Rle (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) den.
L180341
rewrite the current goal using HdenDef (from left to right) at position 1.
L180341
An exact proof term for the current goal is Hup2.
L180342
We prove the intermediate claim HmdenR: minus_SNo den ∈ R.
L180344
An exact proof term for the current goal is (real_minus_SNo den HdenR).
L180344
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo den) den (add_SNo (apply_fun r x) (minus_SNo (apply_fun (u_of r) x))) HmdenR HdenR HrNumR Hlow Hup).
L180347
We prove the intermediate claim HrNumR: apply_fun rNum x ∈ R.
L180349
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun rNum x) HrNumI2).
L180349
We prove the intermediate claim HrNewEq: apply_fun rNew x = div_SNo (apply_fun rNum x) den.
L180351
An exact proof term for the current goal is (compose_div_const_fun_apply A rNum den x HxA HdenR HrNumR).
L180351
rewrite the current goal using HrNewEq (from left to right).
L180352
An exact proof term for the current goal is (div_SNo_closed_interval_scale den (apply_fun rNum x) HdenR HdenPos HrNumI2).
L180353
We prove the intermediate claim HrNew_contI': continuous_map A Ta I (subspace_topology R R_standard_topology I) rNew.
L180355
An exact proof term for the current goal is (continuous_map_range_restrict A Ta R R_standard_topology rNew I HrNew_contR HIcR HrNew_img).
L180355
rewrite the current goal using HTiEq (from left to right).
L180356
An exact proof term for the current goal is HrNew_contI'.
L180357
Let n be given.
L180358
Assume HnO: n ∈ ω.
L180358
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L180359
We prove the intermediate claim HInv_g_R_A: ∀n : set, n ∈ ω → ∀x : set, x ∈ A → apply_fun ((State n) 0) x ∈ R.
(*** Step II invariant on A: each g_n takes values in R ***)
L180364
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ A → apply_fun ((State n) 0) x ∈ R.
L180366
Apply nat_ind to the current goal.
L180367
Let x be given.
L180367
Assume HxA: x ∈ A.
L180367
We will prove apply_fun ((State 0) 0) x ∈ R.
L180368
We prove the intermediate claim HxX: x ∈ X.
L180370
An exact proof term for the current goal is (HAsubX x HxA).
L180370
We prove the intermediate claim H0: State 0 = BaseState.
L180372
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L180372
rewrite the current goal using H0 (from left to right).
L180373
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L180375
rewrite the current goal using HBase (from left to right).
L180376
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L180377
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L180378
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L180380
An exact proof term for the current goal is (Hfung0 x HxX).
L180380
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L180382
Let k be given.
L180382
Assume HkNat: nat_p k.
L180382
Assume IH: ∀x : set, x ∈ A → apply_fun ((State k) 0) x ∈ R.
L180383
Let x be given.
L180384
Assume HxA: x ∈ A.
L180384
We will prove apply_fun ((State (ordsucc k)) 0) x ∈ R.
L180385
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L180387
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L180387
rewrite the current goal using HS (from left to right).
L180388
We prove the intermediate claim HkO: k ∈ ω.
L180390
An exact proof term for the current goal is (nat_p_omega k HkNat).
L180390
We prove the intermediate claim HxX: x ∈ X.
L180392
An exact proof term for the current goal is (HAsubX x HxA).
L180392
Set st to be the term State k.
L180393
Set g to be the term st 0.
L180394
Set rpack to be the term st 1.
L180395
Set r to be the term rpack 0.
L180396
Set c to be the term rpack 1.
L180397
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L180400
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L180405
Set cNew to be the term mul_SNo c den.
L180406
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L180408
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L180410
rewrite the current goal using Hdef (from left to right).
L180410
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L180411
rewrite the current goal using HgStep (from left to right).
L180412
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L180414
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L180416
rewrite the current goal using HgEq (from left to right).
L180416
An exact proof term for the current goal is (IH x HxA).
L180417
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L180419
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L180421
rewrite the current goal using HrEq (from left to right).
L180421
An exact proof term for the current goal is (HInv_r_contI k HkO).
L180422
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L180429
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L180429
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L180431
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L180441
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L180443
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L180443
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L180445
An exact proof term for the current goal is (Hu_fun x HxX).
L180445
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L180447
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L180447
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L180449
We prove the intermediate claim HcR: c ∈ R.
L180451
rewrite the current goal using HcEq (from left to right).
L180451
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L180455
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L180459
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L180462
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L180462
rewrite the current goal using Hcomp (from left to right).
L180463
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L180464
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L180466
rewrite the current goal using HcorrEq (from left to right).
L180466
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L180467
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L180472
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L180472
rewrite the current goal using HgNewEval (from left to right).
L180473
An exact proof term for the current goal is (real_add_SNo (apply_fun g x) HgxR (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x) HcorrR).
L180475
Let n be given.
L180476
Assume HnO: n ∈ ω.
L180476
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L180477
We prove the intermediate claim HInv_residual_identity_A: ∀n : set, n ∈ ω → ∀x : set, x ∈ A → add_SNo (apply_fun ((State n) 0) x) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = apply_fun f x.
(*** Step II invariant on A: f(x) = g_n(x) + c_n times r_n(x) ***)
L180485
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ A → add_SNo (apply_fun ((State n) 0) x) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = apply_fun f x.
L180489
Apply nat_ind to the current goal.
L180490
Let x be given.
L180490
Assume HxA: x ∈ A.
L180490
We will prove add_SNo (apply_fun ((State 0) 0) x) (mul_SNo (apply_fun (((State 0) 1) 0) x) (((State 0) 1) 1)) = apply_fun f x.
L180493
We prove the intermediate claim HxX: x ∈ X.
L180495
An exact proof term for the current goal is (HAsubX x HxA).
L180495
We prove the intermediate claim H0: State 0 = BaseState.
L180497
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L180497
rewrite the current goal using H0 (from left to right).
L180498
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L180500
rewrite the current goal using HBase (from left to right).
L180501
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right) at position 1.
L180502
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L180503
rewrite the current goal using (tuple_2_0_eq f1s den) (from left to right) at position 1.
L180504
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L180505
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
L180506
We prove the intermediate claim Hg0g_app: apply_fun g0g x = apply_fun g0 x.
L180508
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
Use reflexivity.
L180509
rewrite the current goal using Hg0g_app (from left to right) at position 1.
L180510
We prove the intermediate claim Hf1s_app: apply_fun f1s x = div_SNo (apply_fun f1 x) den.
L180512
An exact proof term for the current goal is (Hf1s_apply x HxA).
L180512
rewrite the current goal using Hf1s_app (from left to right) at position 1.
L180513
We prove the intermediate claim Hf1xI2: apply_fun f1 x ∈ I2.
L180515
An exact proof term for the current goal is (Hf1_range x HxA).
L180515
We prove the intermediate claim Hf1xI2': apply_fun f1 x ∈ closed_interval (minus_SNo den) den.
L180517
We prove the intermediate claim HI2def: I2 = closed_interval (minus_SNo den) den.
Use reflexivity.
L180518
rewrite the current goal using HI2def (from left to right).
L180519
An exact proof term for the current goal is Hf1xI2.
L180520
We prove the intermediate claim Hf1xR: apply_fun f1 x ∈ R.
L180522
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo den) den (apply_fun f1 x) Hf1xI2').
L180522
We prove the intermediate claim Hf1xS: SNo (apply_fun f1 x).
L180524
An exact proof term for the current goal is (real_SNo (apply_fun f1 x) Hf1xR).
L180524
We prove the intermediate claim HmulDiv: mul_SNo (div_SNo (apply_fun f1 x) den) den = apply_fun f1 x.
L180526
An exact proof term for the current goal is (mul_div_SNo_invL (apply_fun f1 x) den Hf1xS H23S H23ne0).
L180526
rewrite the current goal using HmulDiv (from left to right) at position 1.
L180527
rewrite the current goal using (Hf1_apply x HxA) (from left to right) at position 1.
L180528
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L180530
An exact proof term for the current goal is (Hf_R x HxA).
L180530
We prove the intermediate claim HfxS: SNo (apply_fun f x).
L180532
An exact proof term for the current goal is (real_SNo (apply_fun f x) HfxR).
L180532
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L180534
An exact proof term for the current goal is (Hfung0 x HxX).
L180534
We prove the intermediate claim Hg0R: apply_fun g0 x ∈ R.
L180536
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L180536
We prove the intermediate claim Hg0S: SNo (apply_fun g0 x).
L180538
An exact proof term for the current goal is (real_SNo (apply_fun g0 x) Hg0R).
L180538
rewrite the current goal using (add_SNo_assoc (apply_fun g0 x) (apply_fun f x) (minus_SNo (apply_fun g0 x)) Hg0S HfxS (SNo_minus_SNo (apply_fun g0 x) Hg0S)) (from left to right) at position 1.
L180540
rewrite the current goal using (add_SNo_com (apply_fun g0 x) (apply_fun f x) Hg0S HfxS) (from left to right) at position 1.
L180541
rewrite the current goal using (add_SNo_assoc (apply_fun f x) (apply_fun g0 x) (minus_SNo (apply_fun g0 x)) HfxS Hg0S (SNo_minus_SNo (apply_fun g0 x) Hg0S)) (from right to left) at position 1.
L180543
rewrite the current goal using (add_SNo_minus_SNo_rinv (apply_fun g0 x) Hg0S) (from left to right) at position 1.
L180544
rewrite the current goal using (add_SNo_0R (apply_fun f x) HfxS) (from left to right) at position 1.
Use reflexivity.
L180547
Let k be given.
L180547
Assume HkNat: nat_p k.
L180547
Assume IH: ∀x : set, x ∈ A → add_SNo (apply_fun ((State k) 0) x) (mul_SNo (apply_fun (((State k) 1) 0) x) (((State k) 1) 1)) = apply_fun f x.
L180551
Let x be given.
L180552
Assume HxA: x ∈ A.
L180552
We will prove add_SNo (apply_fun ((State (ordsucc k)) 0) x) (mul_SNo (apply_fun (((State (ordsucc k)) 1) 0) x) (((State (ordsucc k)) 1) 1)) = apply_fun f x.
L180555
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L180557
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L180557
rewrite the current goal using HS (from left to right).
L180558
rewrite the current goal using (IH x HxA) (from right to left).
L180559
We prove the intermediate claim HkO: k ∈ ω.
L180561
An exact proof term for the current goal is (nat_p_omega k HkNat).
L180561
We prove the intermediate claim HxX: x ∈ X.
L180563
An exact proof term for the current goal is (HAsubX x HxA).
L180563
Set st to be the term State k.
L180564
Set g to be the term st 0.
L180565
Set rpack to be the term st 1.
L180566
Set r to be the term rpack 0.
L180567
Set c to be the term rpack 1.
L180568
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L180571
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L180576
Set cNew to be the term mul_SNo c den.
L180577
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L180579
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L180581
rewrite the current goal using Hdef (from left to right).
L180581
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L180582
We prove the intermediate claim HrpackStep: (StepState k st) 1 = (rNew,cNew).
L180584
rewrite the current goal using Hdef (from left to right).
L180584
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L180585
We prove the intermediate claim HrStep: ((StepState k st) 1) 0 = rNew.
L180587
rewrite the current goal using HrpackStep (from left to right).
L180587
An exact proof term for the current goal is (tuple_2_0_eq rNew cNew).
L180588
We prove the intermediate claim HcStep: ((StepState k st) 1) 1 = cNew.
L180590
rewrite the current goal using HrpackStep (from left to right).
L180590
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L180591
rewrite the current goal using HgStep (from left to right) at position 1.
L180592
rewrite the current goal using HrStep (from left to right) at position 1.
L180593
rewrite the current goal using HcStep (from left to right) at position 1.
L180594
Set gx to be the term apply_fun g x.
L180595
Set rx to be the term apply_fun r x.
L180596
Set ux to be the term apply_fun (u_of r) x.
L180597
We prove the intermediate claim HgxR: gx ∈ R.
L180599
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L180600
rewrite the current goal using HgEq (from left to right).
L180601
An exact proof term for the current goal is (HInv_g_R_A k HkO x HxA).
L180602
We prove the intermediate claim HgxS: SNo gx.
L180604
An exact proof term for the current goal is (real_SNo gx HgxR).
L180604
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L180606
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L180608
rewrite the current goal using HrEq (from left to right).
L180608
An exact proof term for the current goal is (HInv_r_contI k HkO).
L180609
We prove the intermediate claim Hr_fun: function_on r A I.
L180611
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti r Hr_contI).
L180611
We prove the intermediate claim HrxI: rx ∈ I.
L180613
An exact proof term for the current goal is (Hr_fun x HxA).
L180613
We prove the intermediate claim HrxR: rx ∈ R.
L180615
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 rx HrxI).
L180615
We prove the intermediate claim HrxS: SNo rx.
L180617
An exact proof term for the current goal is (real_SNo rx HrxR).
L180617
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L180624
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L180624
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L180626
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L180636
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L180638
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L180638
We prove the intermediate claim HuxI0: ux ∈ I0.
L180640
An exact proof term for the current goal is (Hu_fun x HxX).
L180640
We prove the intermediate claim HuxR: ux ∈ R.
L180642
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third ux HuxI0).
L180642
We prove the intermediate claim HuxS: SNo ux.
L180644
An exact proof term for the current goal is (real_SNo ux HuxR).
L180644
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L180646
We prove the intermediate claim HcR: c ∈ R.
L180648
rewrite the current goal using HcEq (from left to right).
L180648
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L180652
We prove the intermediate claim HcS: SNo c.
L180654
An exact proof term for the current goal is (real_SNo c HcR).
L180654
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo ux c.
L180657
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) ux.
L180660
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L180660
rewrite the current goal using Hcomp (from left to right).
L180661
An exact proof term for the current goal is (mul_const_fun_apply c ux HcR HuxR).
L180662
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L180664
rewrite the current goal using HcorrEq (from left to right).
L180664
An exact proof term for the current goal is (real_mul_SNo ux HuxR c HcR).
L180665
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo gx (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L180669
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L180669
rewrite the current goal using HgNewEval (from left to right).
L180670
rewrite the current goal using HcorrEq (from left to right) at position 1.
L180671
Set uxc to be the term mul_SNo ux c.
L180672
We prove the intermediate claim HuxcR: uxc ∈ R.
L180674
An exact proof term for the current goal is (real_mul_SNo ux HuxR c HcR).
L180674
We prove the intermediate claim HuxcS: SNo uxc.
L180676
An exact proof term for the current goal is (real_SNo uxc HuxcR).
L180676
Set rNum to be the term compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R.
L180677
We prove the intermediate claim HrNumEval: apply_fun rNum x = add_SNo rx (minus_SNo ux).
L180679
An exact proof term for the current goal is (add_of_pair_map_neg_apply A r (u_of r) x HxA HrxR HuxR).
L180679
We prove the intermediate claim HrNumR: apply_fun rNum x ∈ R.
L180681
rewrite the current goal using HrNumEval (from left to right).
L180681
An exact proof term for the current goal is (real_add_SNo rx HrxR (minus_SNo ux) (real_minus_SNo ux HuxR)).
L180682
We prove the intermediate claim HrNewEq: apply_fun rNew x = div_SNo (apply_fun rNum x) den.
L180684
An exact proof term for the current goal is (compose_div_const_fun_apply A rNum den x HxA HdenR HrNumR).
L180684
rewrite the current goal using HrNewEq (from left to right).
L180685
rewrite the current goal using HrNumEval (from left to right) at position 1.
L180686
Set num to be the term add_SNo rx (minus_SNo ux).
L180687
We prove the intermediate claim HnumDef: num = add_SNo rx (minus_SNo ux).
Use reflexivity.
L180689
rewrite the current goal using HnumDef (from right to left) at position 1.
L180690
We prove the intermediate claim HnumR: num ∈ R.
L180692
An exact proof term for the current goal is (real_add_SNo rx HrxR (minus_SNo ux) (real_minus_SNo ux HuxR)).
L180692
We prove the intermediate claim HnumS: SNo num.
L180694
An exact proof term for the current goal is (real_SNo num HnumR).
L180694
We prove the intermediate claim HdenS: SNo den.
L180696
An exact proof term for the current goal is (real_SNo den HdenR).
L180696
We prove the intermediate claim HdivR: div_SNo num den ∈ R.
L180698
An exact proof term for the current goal is (real_div_SNo num HnumR den HdenR).
L180698
We prove the intermediate claim HdivS: SNo (div_SNo num den).
L180700
An exact proof term for the current goal is (real_SNo (div_SNo num den) HdivR).
L180700
We prove the intermediate claim HcNewDef: cNew = mul_SNo c den.
Use reflexivity.
(*** reduce the correction term algebraically ***)
L180703
rewrite the current goal using HcNewDef (from left to right).
L180704
We prove the intermediate claim HmulAssoc: mul_SNo (div_SNo num den) (mul_SNo c den) = mul_SNo (mul_SNo (div_SNo num den) c) den.
L180707
An exact proof term for the current goal is (mul_SNo_assoc (div_SNo num den) c den HdivS HcS HdenS).
L180707
rewrite the current goal using HmulAssoc (from left to right).
L180708
We prove the intermediate claim HmulSwap: mul_SNo (mul_SNo (div_SNo num den) c) den = mul_SNo (mul_SNo (div_SNo num den) den) c.
L180711
An exact proof term for the current goal is (mul_SNo_com_3b_1_2 (div_SNo num den) c den HdivS HcS HdenS).
L180711
rewrite the current goal using HmulSwap (from left to right).
L180712
We prove the intermediate claim Hdenne0: den ≠ 0.
L180714
An exact proof term for the current goal is H23ne0.
L180714
We prove the intermediate claim Hcancel: mul_SNo (div_SNo num den) den = num.
L180716
An exact proof term for the current goal is (mul_div_SNo_invL num den HnumS HdenS Hdenne0).
L180716
rewrite the current goal using Hcancel (from left to right).
L180717
We prove the intermediate claim HnumEq: num = add_SNo rx (minus_SNo ux).
Use reflexivity.
L180719
rewrite the current goal using HnumEq (from left to right).
L180720
We prove the intermediate claim HuxcDef: uxc = mul_SNo ux c.
Use reflexivity.
L180722
rewrite the current goal using HuxcDef (from left to right) at position 1.
L180723
An exact proof term for the current goal is (Tietze_stepII_algebra_tail gx rx ux c HgxS HrxS HuxS HcS).
L180724
Let n be given.
L180725
Assume HnO: n ∈ ω.
L180725
Let x be given.
L180726
Assume HxA: x ∈ A.
L180726
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO) x HxA).
L180727
We prove the intermediate claim HInv_g_FS: ∀n : set, n ∈ ω → (State n) 0 ∈ function_space X R.
(*** The remaining proof obligations are split so they can be refined independently later. ***)
(*** helper: State n yields a real-valued map on X (in function_space X R) ***)
L180731
We prove the intermediate claim Hfun_g0: function_on g0 X R.
L180732
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR).
L180732
We prove the intermediate claim Hg0g_FS: g0g ∈ function_space X R.
L180734
Set gg to be the term (λx : set ⇒ apply_fun g0 x).
L180734
We prove the intermediate claim Hgg: ∀x : set, x ∈ X → gg x ∈ R.
L180736
Let x be given.
L180736
Assume HxX: x ∈ X.
L180736
An exact proof term for the current goal is (Hfun_g0 x HxX).
L180737
We prove the intermediate claim Hg0g_def: g0g = graph X gg.
Use reflexivity.
L180739
rewrite the current goal using Hg0g_def (from left to right).
L180740
An exact proof term for the current goal is (graph_in_function_space X R gg Hgg).
L180741
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → (State n) 0 ∈ function_space X R.
L180743
Apply nat_ind to the current goal.
L180744
We will prove (State 0) 0 ∈ function_space X R.
L180745
We prove the intermediate claim H0: State 0 = BaseState.
(*** base case ***)
L180747
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L180747
rewrite the current goal using H0 (from left to right).
L180748
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L180750
rewrite the current goal using HBase (from left to right).
L180751
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L180752
An exact proof term for the current goal is Hg0g_FS.
L180754
Let n be given.
L180754
Assume HnNat: nat_p n.
L180754
Assume IH: (State n) 0 ∈ function_space X R.
L180755
We will prove (State (ordsucc n)) 0 ∈ function_space X R.
L180756
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L180758
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L180758
rewrite the current goal using HS (from left to right).
L180759
Set st to be the term State n.
L180760
Set g to be the term st 0.
L180761
Set rpack to be the term st 1.
L180762
Set r to be the term rpack 0.
L180763
Set c to be the term rpack 1.
L180764
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L180767
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L180772
Set cNew to be the term mul_SNo c den.
L180773
We prove the intermediate claim HStep0: (StepState n st) 0 = gNew.
L180775
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L180776
rewrite the current goal using Hdef (from left to right).
L180777
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L180778
rewrite the current goal using HStep0 (from left to right).
L180779
We prove the intermediate claim Hg_on: function_on g X R.
L180781
An exact proof term for the current goal is (function_on_of_function_space g X R IH).
L180781
We prove the intermediate claim HnO: n ∈ ω.
L180783
An exact proof term for the current goal is (nat_p_omega n HnNat).
L180783
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L180785
An exact proof term for the current goal is (HInv_cpos n HnO).
L180785
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L180787
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L180789
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L180789
We prove the intermediate claim HcR: c ∈ R.
L180791
rewrite the current goal using Hc_eq (from left to right).
L180791
An exact proof term for the current goal is HcR0.
L180791
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L180793
An exact proof term for the current goal is (HInv_r_contI n HnO).
L180793
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L180800
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L180800
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L180805
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L180811
We prove the intermediate claim Hu_cont: continuous_map X Tx I0 T0 (u_of r).
L180813
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L180817
We prove the intermediate claim Hu_fun_I0: function_on (u_of r) X I0.
L180819
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_cont).
L180819
We prove the intermediate claim HI0SubR: I0 ⊆ R.
L180821
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L180821
We prove the intermediate claim Hu_fun_R: function_on (u_of r) X R.
L180823
Let x be given.
L180823
Assume HxX: x ∈ X.
L180823
An exact proof term for the current goal is (HI0SubR (apply_fun (u_of r) x) (Hu_fun_I0 x HxX)).
L180824
We prove the intermediate claim HmulFS: mul_const_fun c ∈ function_space R R.
L180826
An exact proof term for the current goal is (mul_const_fun_in_function_space c HcR).
L180826
We prove the intermediate claim Hmul_on_R: function_on (mul_const_fun c) R R.
L180828
An exact proof term for the current goal is (function_on_of_function_space (mul_const_fun c) R R HmulFS).
L180828
We prove the intermediate claim Hmul_on_I0: function_on (mul_const_fun c) I0 R.
L180830
An exact proof term for the current goal is (function_on_restrict_dom (mul_const_fun c) R I0 R Hmul_on_R HI0SubR).
L180830
We prove the intermediate claim Hh_on: function_on (compose_fun X (u_of r) (mul_const_fun c)) X R.
L180832
An exact proof term for the current goal is (function_on_compose_fun X I0 R (u_of r) (mul_const_fun c) Hu_fun_I0 Hmul_on_I0).
L180832
We prove the intermediate claim Hpair_FS: pair_map X g (compose_fun X (u_of r) (mul_const_fun c)) ∈ function_space X (setprod R R).
L180834
An exact proof term for the current goal is (pair_map_in_function_space X R R g (compose_fun X (u_of r) (mul_const_fun c)) Hg_on Hh_on).
L180834
We prove the intermediate claim Hpair_on: function_on (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) X (setprod R R).
L180836
An exact proof term for the current goal is (function_on_of_function_space (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) X (setprod R R) Hpair_FS).
L180836
We prove the intermediate claim HaddFS: add_fun_R ∈ function_space (setprod R R) R.
L180838
An exact proof term for the current goal is add_fun_R_in_function_space.
L180838
We prove the intermediate claim Hadd_on: function_on add_fun_R (setprod R R) R.
L180840
An exact proof term for the current goal is (function_on_of_function_space add_fun_R (setprod R R) R HaddFS).
L180840
An exact proof term for the current goal is (compose_fun_in_function_space X (setprod R R) R (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R Hpair_on Hadd_on).
L180843
Let n be given.
L180844
Assume HnO: n ∈ ω.
L180844
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L180845
Apply andI to the current goal.
L180847
We will prove (((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x))).
L180855
Apply andI to the current goal.
L180857
We will prove ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)).
L180860
Apply andI to the current goal.
L180862
We will prove (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))).
L180863
Apply andI to the current goal.
L180865
Let n be given.
L180866
Assume HnO: n ∈ ω.
L180866
We will prove apply_fun fn n ∈ function_space X R.
(*** fn n lies in function_space X R (invariant on State n) ***)
L180867
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L180869
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L180870
An exact proof term for the current goal is (HInv_g_FS n HnO).
L180872
We prove the intermediate claim HInv_g_cont: ∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology ((State n) 0).
(*** continuity of each fn n (invariant on State n) ***)
L180874
We prove the intermediate claim Hg0g_cont: continuous_map X Tx R R_standard_topology g0g.
L180875
We prove the intermediate claim HTx': topology_on X Tx.
L180876
An exact proof term for the current goal is (continuous_map_topology_dom X Tx R R_standard_topology g0 Hg0contR).
L180876
We prove the intermediate claim HTR: topology_on R R_standard_topology.
L180878
An exact proof term for the current goal is (continuous_map_topology_cod X Tx R R_standard_topology g0 Hg0contR).
L180878
We prove the intermediate claim Hfun_g0: function_on g0 X R.
L180880
An exact proof term for the current goal is (continuous_map_function_on X Tx R R_standard_topology g0 Hg0contR).
L180880
We prove the intermediate claim Hfun_g0g: function_on g0g X R.
L180882
Let x be given.
L180882
Assume HxX: x ∈ X.
L180882
We will prove apply_fun g0g x ∈ R.
L180883
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right) at position 1.
L180884
An exact proof term for the current goal is (Hfun_g0 x HxX).
L180885
We prove the intermediate claim Hpre_g0g: ∀V : set, V ∈ R_standard_topology → preimage_of X g0g V ∈ Tx.
L180887
Let V be given.
L180887
Assume HV: V ∈ R_standard_topology.
L180887
We prove the intermediate claim Hpre_g0: preimage_of X g0 V ∈ Tx.
L180889
An exact proof term for the current goal is (continuous_map_preimage X Tx R R_standard_topology g0 Hg0contR V HV).
L180889
We prove the intermediate claim Heq: preimage_of X g0g V = preimage_of X g0 V.
L180891
Apply set_ext to the current goal.
L180892
Let x be given.
L180892
Assume Hx: x ∈ preimage_of X g0g V.
L180892
We will prove x ∈ preimage_of X g0 V.
L180893
We prove the intermediate claim HxX: x ∈ X.
L180895
An exact proof term for the current goal is (SepE1 X (λy : set ⇒ apply_fun g0g y ∈ V) x Hx).
L180895
We prove the intermediate claim Himg: apply_fun g0g x ∈ V.
L180897
An exact proof term for the current goal is (SepE2 X (λy : set ⇒ apply_fun g0g y ∈ V) x Hx).
L180897
We prove the intermediate claim Himg': apply_fun g0 x ∈ V.
L180899
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from right to left) at position 1.
L180899
An exact proof term for the current goal is Himg.
L180900
An exact proof term for the current goal is (SepI X (λy : set ⇒ apply_fun g0 y ∈ V) x HxX Himg').
L180902
Let x be given.
L180902
Assume Hx: x ∈ preimage_of X g0 V.
L180902
We will prove x ∈ preimage_of X g0g V.
L180903
We prove the intermediate claim HxX: x ∈ X.
L180905
An exact proof term for the current goal is (SepE1 X (λy : set ⇒ apply_fun g0 y ∈ V) x Hx).
L180905
We prove the intermediate claim Himg: apply_fun g0 x ∈ V.
L180907
An exact proof term for the current goal is (SepE2 X (λy : set ⇒ apply_fun g0 y ∈ V) x Hx).
L180907
We prove the intermediate claim Himg': apply_fun g0g x ∈ V.
L180909
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right) at position 1.
L180909
An exact proof term for the current goal is Himg.
L180910
An exact proof term for the current goal is (SepI X (λy : set ⇒ apply_fun g0g y ∈ V) x HxX Himg').
L180911
rewrite the current goal using Heq (from left to right).
L180912
An exact proof term for the current goal is Hpre_g0.
L180913
We will prove continuous_map X Tx R R_standard_topology g0g.
L180914
We will prove topology_on X Tx ∧ topology_on R R_standard_topology ∧ function_on g0g X R ∧ ∀V : set, V ∈ R_standard_topology → preimage_of X g0g V ∈ Tx.
L180917
We prove the intermediate claim Htop: topology_on X Tx ∧ topology_on R R_standard_topology.
L180919
Apply andI to the current goal.
L180919
An exact proof term for the current goal is HTx'.
L180919
An exact proof term for the current goal is HTR.
L180919
We prove the intermediate claim Htopfun: (topology_on X Tx ∧ topology_on R R_standard_topology) ∧ function_on g0g X R.
L180921
Apply andI to the current goal.
L180921
An exact proof term for the current goal is Htop.
L180921
An exact proof term for the current goal is Hfun_g0g.
L180921
Apply andI to the current goal.
L180923
An exact proof term for the current goal is Htopfun.
L180924
An exact proof term for the current goal is Hpre_g0g.
L180924
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → continuous_map X Tx R R_standard_topology ((State n) 0).
L180926
Apply nat_ind to the current goal.
L180927
We will prove continuous_map X Tx R R_standard_topology ((State 0) 0).
L180927
We prove the intermediate claim H0: State 0 = BaseState.
L180929
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L180929
rewrite the current goal using H0 (from left to right).
L180930
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L180932
rewrite the current goal using HBase (from left to right).
L180933
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L180934
An exact proof term for the current goal is Hg0g_cont.
L180936
Let n be given.
L180936
Assume HnNat: nat_p n.
L180936
Assume IH: continuous_map X Tx R R_standard_topology ((State n) 0).
L180937
We will prove continuous_map X Tx R R_standard_topology ((State (ordsucc n)) 0).
L180938
We prove the intermediate claim HS: State (ordsucc n) = StepState n (State n).
L180940
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L180940
rewrite the current goal using HS (from left to right).
L180941
Set st to be the term State n.
L180942
Set g to be the term st 0.
L180943
Set rpack to be the term st 1.
L180944
Set r to be the term rpack 0.
L180945
Set c to be the term rpack 1.
L180946
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L180949
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L180954
Set cNew to be the term mul_SNo c den.
L180955
We prove the intermediate claim HStep0: (StepState n st) 0 = gNew.
L180957
We prove the intermediate claim Hdef: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L180958
rewrite the current goal using Hdef (from left to right).
L180959
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L180960
rewrite the current goal using HStep0 (from left to right).
L180961
We prove the intermediate claim Hg_cont: continuous_map X Tx R R_standard_topology g.
L180963
An exact proof term for the current goal is IH.
L180963
We prove the intermediate claim HnO: n ∈ ω.
L180965
An exact proof term for the current goal is (nat_p_omega n HnNat).
L180965
We prove the intermediate claim Hcpack: ((State n) 1) 1 ∈ R ∧ 0 < ((State n) 1) 1.
L180967
An exact proof term for the current goal is (HInv_cpos n HnO).
L180967
We prove the intermediate claim Hc_eq: c = ((State n) 1) 1.
Use reflexivity.
L180969
We prove the intermediate claim HcR0: ((State n) 1) 1 ∈ R.
L180971
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L180971
We prove the intermediate claim Hcpos0: 0 < ((State n) 1) 1.
L180973
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) Hcpack).
L180973
We prove the intermediate claim HcR: c ∈ R.
L180975
rewrite the current goal using Hc_eq (from left to right).
L180975
An exact proof term for the current goal is HcR0.
L180975
We prove the intermediate claim Hcpos: 0 < c.
L180977
rewrite the current goal using Hc_eq (from left to right).
L180977
An exact proof term for the current goal is Hcpos0.
L180977
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L180979
An exact proof term for the current goal is (HInv_r_contI n HnO).
L180979
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third).
L180986
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L180986
We prove the intermediate claim Hu_left: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third).
L180991
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r) ∧ (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third)) (∀x : set, x ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x = one_third) Hu_pack).
L180997
We prove the intermediate claim Hu_cont: continuous_map X Tx I0 T0 (u_of r).
L180999
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x : set, x ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x = minus_SNo one_third) Hu_left).
L181003
We prove the intermediate claim HI0SubR: I0 ⊆ R.
L181005
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third).
L181005
We prove the intermediate claim HT0: T0 = subspace_topology R R_standard_topology I0.
Use reflexivity.
L181007
We prove the intermediate claim Hmul_contR: continuous_map R R_standard_topology R R_standard_topology (mul_const_fun c).
L181009
An exact proof term for the current goal is (mul_const_fun_continuous_pos c HcR Hcpos).
L181009
We prove the intermediate claim Hmul_contI0: continuous_map I0 T0 R R_standard_topology (mul_const_fun c).
L181011
rewrite the current goal using HT0 (from left to right).
L181011
An exact proof term for the current goal is (continuous_on_subspace R R_standard_topology R R_standard_topology (mul_const_fun c) I0 R_standard_topology_is_topology HI0SubR Hmul_contR).
L181013
We prove the intermediate claim Hh_cont: continuous_map X Tx R R_standard_topology (compose_fun X (u_of r) (mul_const_fun c)).
L181015
An exact proof term for the current goal is (composition_continuous X Tx I0 T0 R R_standard_topology (u_of r) (mul_const_fun c) Hu_cont Hmul_contI0).
L181016
We prove the intermediate claim Hpair_cont: continuous_map X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))).
L181020
An exact proof term for the current goal is (maps_into_products_axiom X Tx R R_standard_topology R R_standard_topology g (compose_fun X (u_of r) (mul_const_fun c)) Hg_cont Hh_cont).
L181022
We prove the intermediate claim Hadd_cont: continuous_map (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology add_fun_R.
L181026
An exact proof term for the current goal is add_fun_R_continuous.
L181026
An exact proof term for the current goal is (composition_continuous X Tx (setprod R R) (product_topology R R_standard_topology R R_standard_topology) R R_standard_topology (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R Hpair_cont Hadd_cont).
L181033
Let n be given.
L181034
Assume HnO: n ∈ ω.
L181034
An exact proof term for the current goal is (HInv_nat n (omega_nat_p n HnO)).
L181035
Let n be given.
L181036
Assume HnO: n ∈ ω.
L181036
We will prove continuous_map X Tx R R_standard_topology (apply_fun fn n).
L181037
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L181039
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L181040
An exact proof term for the current goal is (HInv_g_cont n HnO).
L181042
We prove the intermediate claim HInv_g_I: ∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I.
(*** range-in-I of each fn n (invariant on State n) ***)
L181044
We prove the intermediate claim HInv_nat: ∀n : set, nat_p n → ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I ∧ apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1))).
L181048
Apply nat_ind to the current goal.
L181049
Let x be given.
L181049
Assume HxX: x ∈ X.
L181049
We will prove apply_fun ((State 0) 0) x ∈ I ∧ apply_fun ((State 0) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State 0) 1) 1)) (add_SNo 1 (minus_SNo (((State 0) 1) 1))).
L181052
We prove the intermediate claim H0: State 0 = BaseState.
L181054
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181054
rewrite the current goal using H0 (from left to right).
L181055
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181057
rewrite the current goal using HBase (from left to right).
L181058
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L181059
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L181060
We prove the intermediate claim Hc0Eq: ((g0g,(f1s,den)) 1) 1 = den.
L181062
rewrite the current goal using (tuple_2_1_eq g0g (f1s,den)) (from left to right) at position 1.
L181062
rewrite the current goal using (tuple_2_1_eq f1s den) (from left to right) at position 1.
Use reflexivity.
L181064
Apply andI to the current goal.
L181066
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L181067
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
(*** in I ***)
L181069
An exact proof term for the current goal is (Hfung0 x HxX).
L181069
An exact proof term for the current goal is (closed_interval_minus_one_third_one_third_sub_closed_interval_minus1_1 (apply_fun g0 x) Hg0xI0).
L181071
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX) (from left to right).
L181072
We prove the intermediate claim Hg0xI0: apply_fun g0 x ∈ I0.
(*** in the shrinking interval for c0=den=2/3 ***)
L181074
An exact proof term for the current goal is (Hfung0 x HxX).
L181074
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181076
We prove the intermediate claim HlowEq0: add_SNo (minus_SNo 1) (((g0g,(f1s,den)) 1) 1) = minus_SNo one_third.
L181079
rewrite the current goal using Hc0Eq (from left to right).
L181079
rewrite the current goal using HdenDef (from left to right).
L181080
An exact proof term for the current goal is add_minus1_two_thirds_eq_minus_one_third.
L181081
We prove the intermediate claim HupEq0: add_SNo 1 (minus_SNo (((g0g,(f1s,den)) 1) 1)) = one_third.
L181084
rewrite the current goal using Hc0Eq (from left to right).
L181084
rewrite the current goal using HdenDef (from left to right).
L181085
An exact proof term for the current goal is add_1_minus_two_thirds_eq_one_third.
L181086
rewrite the current goal using HlowEq0 (from left to right).
L181087
rewrite the current goal using HupEq0 (from left to right).
L181088
An exact proof term for the current goal is Hg0xI0.
L181090
Let n be given.
L181090
Assume HnNat: nat_p n.
L181090
Assume IH: ∀x : set, x ∈ X → apply_fun ((State n) 0) x ∈ I ∧ apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1))).
L181094
Let x be given.
L181095
Assume HxX: x ∈ X.
L181095
We will prove apply_fun ((State (ordsucc n)) 0) x ∈ I ∧ apply_fun ((State (ordsucc n)) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State (ordsucc n)) 1) 1)) (add_SNo 1 (minus_SNo (((State (ordsucc n)) 1) 1))).
L181098
We prove the intermediate claim HSnat: State (ordsucc n) = StepState n (State n).
(*** TODO: step case needs the geometric-series bound plus range control of the correction term. ***)
L181101
An exact proof term for the current goal is (nat_primrec_S BaseState StepState n HnNat).
L181101
rewrite the current goal using HSnat (from left to right).
L181102
Set st to be the term State n.
L181103
Set g to be the term st 0.
L181104
Set rpack to be the term st 1.
L181105
Set r to be the term rpack 0.
L181106
Set c to be the term rpack 1.
L181107
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181110
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181115
Set cNew to be the term mul_SNo c den.
L181116
We prove the intermediate claim HdefStep: StepState n st = (gNew,(rNew,cNew)).
Use reflexivity.
L181118
rewrite the current goal using HdefStep (from left to right).
L181119
We prove the intermediate claim HgStep: ((gNew,(rNew,cNew)) 0) = gNew.
L181121
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L181121
rewrite the current goal using HgStep (from left to right).
L181122
We prove the intermediate claim HnO: n ∈ ω.
L181124
An exact proof term for the current goal is (nat_p_omega n HnNat).
L181124
We prove the intermediate claim HrEq: r = ((State n) 1) 0.
Use reflexivity.
L181126
We prove the intermediate claim HcEq: c = ((State n) 1) 1.
Use reflexivity.
L181128
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L181130
rewrite the current goal using HrEq (from left to right).
L181130
An exact proof term for the current goal is (HInv_r_contI n HnO).
L181131
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L181138
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L181138
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L181140
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L181150
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L181152
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L181152
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L181154
An exact proof term for the current goal is (Hu_fun x HxX).
L181154
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L181156
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L181156
We prove the intermediate claim HcR: c ∈ R.
L181158
rewrite the current goal using HcEq (from left to right).
L181158
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L181162
We prove the intermediate claim HgEq: g = (State n) 0.
Use reflexivity.
L181164
We prove the intermediate claim HgxI: apply_fun g x ∈ I.
L181166
rewrite the current goal using HgEq (from left to right).
L181166
An exact proof term for the current goal is (andEL (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (IH x HxX)).
L181171
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L181173
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun g x) HgxI).
L181173
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L181177
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L181180
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX).
L181180
rewrite the current goal using Hcomp (from left to right).
L181181
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L181182
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L181184
rewrite the current goal using HcorrEq (from left to right).
L181184
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L181185
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L181189
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX HgxR HcorrR).
L181189
rewrite the current goal using HgNewEval (from left to right).
L181191
rewrite the current goal using HcorrEq (from left to right).
(*** TODO: finish by bounding the correction term and using interval closure under this controlled addition. ***)
L181192
We prove the intermediate claim Hcpos: 0 < c.
L181194
rewrite the current goal using HcEq (from left to right).
L181194
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L181198
We prove the intermediate claim HyIshrink: apply_fun g x ∈ closed_interval (add_SNo (minus_SNo 1) c) (add_SNo 1 (minus_SNo c)).
L181201
rewrite the current goal using HgEq (from left to right).
L181201
rewrite the current goal using HcEq (from left to right).
L181202
An exact proof term for the current goal is (andER (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (IH x HxX)).
L181207
We prove the intermediate claim HzIthird: mul_SNo (apply_fun (u_of r) x) c ∈ closed_interval (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third).
L181211
We will prove mul_SNo (apply_fun (u_of r) x) c ∈ closed_interval (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third).
L181211
We prove the intermediate claim H13R: one_third ∈ R.
L181213
An exact proof term for the current goal is one_third_in_R.
L181213
We prove the intermediate claim H13S: SNo one_third.
L181215
An exact proof term for the current goal is (real_SNo one_third H13R).
L181215
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L181217
An exact proof term for the current goal is (real_minus_SNo one_third H13R).
L181217
We prove the intermediate claim Hm13S: SNo (minus_SNo one_third).
L181219
An exact proof term for the current goal is (SNo_minus_SNo one_third H13S).
L181219
We prove the intermediate claim HuS: SNo (apply_fun (u_of r) x).
L181221
An exact proof term for the current goal is (real_SNo (apply_fun (u_of r) x) HuxR).
L181221
We prove the intermediate claim HcS: SNo c.
L181223
An exact proof term for the current goal is (real_SNo c HcR).
L181223
We prove the intermediate claim Hbnds0: Rle (minus_SNo one_third) (apply_fun (u_of r) x) ∧ Rle (apply_fun (u_of r) x) one_third.
L181227
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third (apply_fun (u_of r) x) Hm13R H13R HuxI0).
L181228
We prove the intermediate claim HloRle: Rle (minus_SNo one_third) (apply_fun (u_of r) x).
L181230
An exact proof term for the current goal is (andEL (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hbnds0).
L181233
We prove the intermediate claim HhiRle: Rle (apply_fun (u_of r) x) one_third.
L181235
An exact proof term for the current goal is (andER (Rle (minus_SNo one_third) (apply_fun (u_of r) x)) (Rle (apply_fun (u_of r) x) one_third) Hbnds0).
L181238
We prove the intermediate claim Hnot_lt_lo: ¬ ((apply_fun (u_of r) x) < (minus_SNo one_third)).
L181240
Assume Hlt: (apply_fun (u_of r) x) < (minus_SNo one_third).
L181240
We prove the intermediate claim Hrlt: Rlt (apply_fun (u_of r) x) (minus_SNo one_third).
L181242
An exact proof term for the current goal is (RltI (apply_fun (u_of r) x) (minus_SNo one_third) HuxR Hm13R Hlt).
L181242
An exact proof term for the current goal is ((RleE_nlt (minus_SNo one_third) (apply_fun (u_of r) x) HloRle) Hrlt).
L181243
We prove the intermediate claim Hnot_lt_hi: ¬ (one_third < (apply_fun (u_of r) x)).
L181245
Assume Hlt: one_third < (apply_fun (u_of r) x).
L181245
We prove the intermediate claim Hrlt: Rlt one_third (apply_fun (u_of r) x).
L181247
An exact proof term for the current goal is (RltI one_third (apply_fun (u_of r) x) H13R HuxR Hlt).
L181247
An exact proof term for the current goal is ((RleE_nlt (apply_fun (u_of r) x) one_third HhiRle) Hrlt).
L181248
We prove the intermediate claim Hm13_le_u: (minus_SNo one_third) ≤ (apply_fun (u_of r) x).
L181250
Apply (SNoLtLe_or (apply_fun (u_of r) x) (minus_SNo one_third) HuS Hm13S) to the current goal.
L181251
Assume Hlt: (apply_fun (u_of r) x) < (minus_SNo one_third).
L181251
An exact proof term for the current goal is (FalseE (Hnot_lt_lo Hlt) ((minus_SNo one_third) ≤ (apply_fun (u_of r) x))).
L181253
Assume Hle: (minus_SNo one_third) ≤ (apply_fun (u_of r) x).
L181253
An exact proof term for the current goal is Hle.
L181254
We prove the intermediate claim Hu_le_13: (apply_fun (u_of r) x) ≤ one_third.
L181256
Apply (SNoLtLe_or one_third (apply_fun (u_of r) x) H13S HuS) to the current goal.
L181257
Assume Hlt: one_third < (apply_fun (u_of r) x).
L181257
An exact proof term for the current goal is (FalseE (Hnot_lt_hi Hlt) ((apply_fun (u_of r) x) ≤ one_third)).
L181259
Assume Hle: (apply_fun (u_of r) x) ≤ one_third.
L181259
An exact proof term for the current goal is Hle.
L181260
We prove the intermediate claim H0le_c: 0 ≤ c.
L181262
An exact proof term for the current goal is (SNoLtLe 0 c Hcpos).
L181262
We prove the intermediate claim HmulLeHi: mul_SNo (apply_fun (u_of r) x) c ≤ mul_SNo one_third c.
L181264
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (apply_fun (u_of r) x) one_third c HuS H13S HcS H0le_c Hu_le_13).
L181264
We prove the intermediate claim HmulLeLo: mul_SNo (minus_SNo one_third) c ≤ mul_SNo (apply_fun (u_of r) x) c.
L181266
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (minus_SNo one_third) (apply_fun (u_of r) x) c Hm13S HuS HcS H0le_c Hm13_le_u).
L181266
We prove the intermediate claim HmulR: mul_SNo (apply_fun (u_of r) x) c ∈ R.
L181268
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L181268
We prove the intermediate claim Hc13R: mul_SNo c one_third ∈ R.
L181270
An exact proof term for the current goal is (real_mul_SNo c HcR one_third H13R).
L181270
We prove the intermediate claim Hm_c13R: minus_SNo (mul_SNo c one_third) ∈ R.
L181272
An exact proof term for the current goal is (real_minus_SNo (mul_SNo c one_third) Hc13R).
L181272
We prove the intermediate claim HhiEq: mul_SNo one_third c = mul_SNo c one_third.
L181274
An exact proof term for the current goal is (mul_SNo_com one_third c H13S HcS).
L181274
We prove the intermediate claim HloEq: mul_SNo (minus_SNo one_third) c = minus_SNo (mul_SNo c one_third).
L181276
rewrite the current goal using (mul_SNo_minus_distrL one_third c H13S HcS) (from left to right) at position 1.
L181276
rewrite the current goal using HhiEq (from left to right) at position 1.
Use reflexivity.
L181278
We prove the intermediate claim HhiLe': mul_SNo (apply_fun (u_of r) x) c ≤ mul_SNo c one_third.
L181280
rewrite the current goal using HhiEq (from right to left).
L181280
An exact proof term for the current goal is HmulLeHi.
L181281
We prove the intermediate claim HloLe': (minus_SNo (mul_SNo c one_third)) ≤ mul_SNo (apply_fun (u_of r) x) c.
L181283
rewrite the current goal using HloEq (from right to left) at position 1.
L181283
An exact proof term for the current goal is HmulLeLo.
L181284
We prove the intermediate claim HhiRle': Rle (mul_SNo (apply_fun (u_of r) x) c) (mul_SNo c one_third).
L181286
An exact proof term for the current goal is (Rle_of_SNoLe (mul_SNo (apply_fun (u_of r) x) c) (mul_SNo c one_third) HmulR Hc13R HhiLe').
L181286
We prove the intermediate claim HloRle': Rle (minus_SNo (mul_SNo c one_third)) (mul_SNo (apply_fun (u_of r) x) c).
L181288
An exact proof term for the current goal is (Rle_of_SNoLe (minus_SNo (mul_SNo c one_third)) (mul_SNo (apply_fun (u_of r) x) c) Hm_c13R HmulR HloLe').
L181288
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo (mul_SNo c one_third)) (mul_SNo c one_third) (mul_SNo (apply_fun (u_of r) x) c) Hm_c13R Hc13R HmulR HloRle' HhiRle').
L181292
We prove the intermediate claim HsumShrink: add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) ∈ closed_interval (add_SNo (minus_SNo 1) (mul_SNo c two_thirds)) (add_SNo 1 (minus_SNo (mul_SNo c two_thirds))).
L181297
An exact proof term for the current goal is (add_SNo_interval_expand_by_third_stub c (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) HcR Hcpos HyIshrink HzIthird).
L181300
Apply andI to the current goal.
L181302
Set t to be the term mul_SNo c two_thirds.
L181303
We prove the intermediate claim H23R: two_thirds ∈ R.
(*** in I (existing argument via HsumShrink) ***)
L181305
An exact proof term for the current goal is two_thirds_in_R.
L181305
We prove the intermediate claim HtR: t ∈ R.
L181307
An exact proof term for the current goal is (real_mul_SNo c HcR two_thirds H23R).
L181307
We prove the intermediate claim Htpos: 0 < t.
L181309
We prove the intermediate claim H23S: SNo two_thirds.
L181310
An exact proof term for the current goal is (real_SNo two_thirds H23R).
L181310
We prove the intermediate claim HcS: SNo c.
L181312
An exact proof term for the current goal is (real_SNo c HcR).
L181312
An exact proof term for the current goal is (mul_SNo_pos_pos c two_thirds HcS H23S Hcpos two_thirds_pos).
L181313
We prove the intermediate claim H0lt_t: Rlt 0 t.
L181315
An exact proof term for the current goal is (RltI 0 t real_0 HtR Htpos).
L181315
We prove the intermediate claim H0le_t: Rle 0 t.
L181317
An exact proof term for the current goal is (Rlt_implies_Rle 0 t H0lt_t).
L181317
We prove the intermediate claim Hm1R: (minus_SNo 1) ∈ R.
L181319
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L181319
We prove the intermediate claim Hm1S: SNo (minus_SNo 1).
L181321
An exact proof term for the current goal is (SNo_minus_SNo 1 SNo_1).
L181321
We prove the intermediate claim Ha_low: Rle (minus_SNo 1) (add_SNo (minus_SNo 1) t).
L181323
We prove the intermediate claim Htmp: Rle (add_SNo (minus_SNo 1) 0) (add_SNo (minus_SNo 1) t).
L181324
An exact proof term for the current goal is (Rle_add_SNo_2 (minus_SNo 1) 0 t Hm1R real_0 HtR H0le_t).
L181324
rewrite the current goal using (add_SNo_0R (minus_SNo 1) Hm1S) (from right to left) at position 1.
L181325
An exact proof term for the current goal is Htmp.
L181326
We prove the intermediate claim Hneg_t_le0: Rle (minus_SNo t) 0.
L181328
An exact proof term for the current goal is (Rle_minus_nonneg t HtR (RleE_nlt 0 t H0le_t)).
L181328
We prove the intermediate claim Hb_up: Rle (add_SNo 1 (minus_SNo t)) 1.
L181330
We prove the intermediate claim Htmp: Rle (add_SNo 1 (minus_SNo t)) (add_SNo 1 0).
L181331
An exact proof term for the current goal is (Rle_add_SNo_2 1 (minus_SNo t) 0 real_1 (real_minus_SNo t HtR) real_0 Hneg_t_le0).
L181331
We prove the intermediate claim H10Eq: add_SNo 1 0 = 1.
L181333
An exact proof term for the current goal is (add_SNo_0R 1 SNo_1).
L181333
We prove the intermediate claim Hmid: Rle (add_SNo 1 0) 1.
L181335
rewrite the current goal using H10Eq (from left to right) at position 1.
L181335
An exact proof term for the current goal is (Rle_refl 1 real_1).
L181336
An exact proof term for the current goal is (Rle_tra (add_SNo 1 (minus_SNo t)) (add_SNo 1 0) 1 Htmp Hmid).
L181337
We prove the intermediate claim HyR: add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c) ∈ R.
L181340
An exact proof term for the current goal is (closed_interval_sub_R (add_SNo (minus_SNo 1) (mul_SNo c two_thirds)) (add_SNo 1 (minus_SNo (mul_SNo c two_thirds))) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) HsumShrink).
L181343
We prove the intermediate claim Hbnds: Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) ∧ Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t)).
L181347
An exact proof term for the current goal is (closed_interval_bounds (add_SNo (minus_SNo 1) t) (add_SNo 1 (minus_SNo t)) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (real_add_SNo (minus_SNo 1) Hm1R t HtR) (real_add_SNo 1 real_1 (minus_SNo t) (real_minus_SNo t HtR)) HsumShrink).
L181351
We prove the intermediate claim Hlo: Rle (minus_SNo 1) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)).
L181354
An exact proof term for the current goal is (Rle_tra (minus_SNo 1) (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) Ha_low (andEL (Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c))) (Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t))) Hbnds)).
L181360
We prove the intermediate claim Hhi: Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) 1.
L181363
An exact proof term for the current goal is (Rle_tra (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t)) 1 (andER (Rle (add_SNo (minus_SNo 1) t) (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c))) (Rle (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) (add_SNo 1 (minus_SNo t))) Hbnds) Hb_up).
L181369
An exact proof term for the current goal is (closed_intervalI_of_Rle (minus_SNo 1) 1 (add_SNo (apply_fun g x) (mul_SNo (apply_fun (u_of r) x) c)) Hm1R real_1 HyR Hlo Hhi).
L181373
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
(*** in the shrinking interval for the successor scalar ***)
L181375
We prove the intermediate claim HcNewEq: cNew = mul_SNo c two_thirds.
L181377
rewrite the current goal using HdenDef (from left to right).
Use reflexivity.
L181378
We prove the intermediate claim HcPairEq: ((gNew,(rNew,cNew)) 1) 1 = cNew.
L181380
We prove the intermediate claim Hinner: ((gNew,(rNew,cNew)) 1) = (rNew,cNew).
L181381
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181381
rewrite the current goal using Hinner (from left to right) at position 1.
L181382
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L181383
We prove the intermediate claim HcGoalEq: ((gNew,(rNew,cNew)) 1) 1 = mul_SNo c two_thirds.
L181385
rewrite the current goal using HcPairEq (from left to right).
L181385
rewrite the current goal using HcNewEq (from left to right).
Use reflexivity.
L181387
We prove the intermediate claim HlowEq: add_SNo (minus_SNo 1) (((gNew,(rNew,cNew)) 1) 1) = add_SNo (minus_SNo 1) (mul_SNo c two_thirds).
L181391
rewrite the current goal using HcGoalEq (from left to right).
Use reflexivity.
L181392
We prove the intermediate claim HupEq: add_SNo 1 (minus_SNo (((gNew,(rNew,cNew)) 1) 1)) = add_SNo 1 (minus_SNo (mul_SNo c two_thirds)).
L181396
rewrite the current goal using HcGoalEq (from left to right).
Use reflexivity.
L181397
rewrite the current goal using HlowEq (from left to right).
L181398
rewrite the current goal using HupEq (from left to right).
L181399
An exact proof term for the current goal is HsumShrink.
L181400
Let n be given.
L181401
Assume HnO: n ∈ ω.
L181401
Let x be given.
L181402
Assume HxX: x ∈ X.
L181402
An exact proof term for the current goal is (andEL (apply_fun ((State n) 0) x ∈ I) (apply_fun ((State n) 0) x ∈ closed_interval (add_SNo (minus_SNo 1) (((State n) 1) 1)) (add_SNo 1 (minus_SNo (((State n) 1) 1)))) (HInv_nat n (omega_nat_p n HnO) x HxX)).
L181407
Let n be given.
L181408
Assume HnO: n ∈ ω.
L181408
Let x be given.
L181409
Assume HxX: x ∈ X.
L181409
We will prove apply_fun (apply_fun fn n) x ∈ I.
L181410
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L181412
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right).
L181413
An exact proof term for the current goal is (HInv_g_I n HnO x HxX).
L181415
Let x be given.
L181416
Assume HxA: x ∈ A.
L181416
We will prove converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x).
(*** pointwise convergence on A (to be proved from residual convergence / geometric tail bounds) ***)
L181419
Set seq1 to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L181420
Set seq2 to be the term graph ω (λn : set ⇒ apply_fun ((State n) 0) x).
L181421
We prove the intermediate claim HseqEq: seq1 = seq2.
L181423
Apply set_ext to the current goal.
L181424
Let p be given.
L181424
Assume Hp: p ∈ seq1.
L181424
We will prove p ∈ seq2.
L181425
Apply (ReplE_impred ω (λn0 : set ⇒ (n0,apply_fun (apply_fun fn n0) x)) p Hp (p ∈ seq2)) to the current goal.
L181426
Let n be given.
L181427
Assume HnO: n ∈ ω.
L181427
Assume HpEq: p = (n,apply_fun (apply_fun fn n) x).
L181428
rewrite the current goal using HpEq (from left to right).
L181429
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L181431
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from left to right) at position 1.
L181432
An exact proof term for the current goal is (ReplI ω (λn0 : set ⇒ (n0,apply_fun ((State n0) 0) x)) n HnO).
L181434
Let p be given.
L181434
Assume Hp: p ∈ seq2.
L181434
We will prove p ∈ seq1.
L181435
Apply (ReplE_impred ω (λn0 : set ⇒ (n0,apply_fun ((State n0) 0) x)) p Hp (p ∈ seq1)) to the current goal.
L181436
Let n be given.
L181437
Assume HnO: n ∈ ω.
L181437
Assume HpEq: p = (n,apply_fun ((State n) 0) x).
L181438
rewrite the current goal using HpEq (from left to right).
L181439
We prove the intermediate claim Hdef: fn = graph ω (λn0 : set ⇒ (State n0) 0).
Use reflexivity.
L181441
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λn0 : set ⇒ (State n0) 0) n Hdef HnO) (from right to left) at position 1.
L181442
An exact proof term for the current goal is (ReplI ω (λn0 : set ⇒ (n0,apply_fun (apply_fun fn n0) x)) n HnO).
L181443
We prove the intermediate claim Hseq1def: seq1 = (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)).
Use reflexivity.
L181446
rewrite the current goal using Hseq1def (from right to left).
L181447
rewrite the current goal using HseqEq (from left to right).
L181448
We prove the intermediate claim Hseq2_conv: converges_to R (metric_topology R R_bounded_metric) seq2 (apply_fun f x).
(*** TODO: show seq2 converges to f x in the bounded-metric topology using the residual recursion. ***)
L181454
Set lim to be the term apply_fun f x.
L181455
We prove the intermediate claim HconvM: sequence_converges_metric R R_bounded_metric seq2 lim.
(*** TODO: derive the usual bound |(State n)0 x - f x| <= ((State n)1)1 on A and use ((State n)1)1 -> 0. ***)
L181457
We will prove metric_on R R_bounded_metric ∧ sequence_on seq2 R ∧ lim ∈ R ∧ ∀eps : set, eps ∈ R ∧ Rlt 0 eps → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L181461
We prove the intermediate claim Hseq2On: sequence_on seq2 R.
L181463
We will prove sequence_on seq2 R.
L181463
Let n be given.
L181464
Assume HnO: n ∈ ω.
L181464
We will prove apply_fun seq2 n ∈ R.
L181465
We prove the intermediate claim HxX: x ∈ X.
L181467
An exact proof term for the current goal is (HAsubX x HxA).
L181467
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L181469
rewrite the current goal using Hseq2def (from left to right).
L181470
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
L181471
We prove the intermediate claim HgR_nat: ∀k : set, nat_p k → apply_fun ((State k) 0) x ∈ R.
L181473
Apply nat_ind to the current goal.
L181474
We will prove apply_fun ((State 0) 0) x ∈ R.
L181474
We prove the intermediate claim H0: State 0 = BaseState.
L181476
An exact proof term for the current goal is (nat_primrec_0 BaseState StepState).
L181476
rewrite the current goal using H0 (from left to right).
L181477
We prove the intermediate claim HBase: BaseState = (g0g,(f1s,den)).
Use reflexivity.
L181479
rewrite the current goal using HBase (from left to right).
L181480
rewrite the current goal using (tuple_2_0_eq g0g (f1s,den)) (from left to right).
L181481
We prove the intermediate claim HxX0: x ∈ X.
L181483
An exact proof term for the current goal is (HAsubX x HxA).
L181483
rewrite the current goal using (apply_fun_graph X (λx0 : set ⇒ apply_fun g0 x0) x HxX0) (from left to right).
L181484
We prove the intermediate claim Hg0I0: apply_fun g0 x ∈ I0.
L181486
An exact proof term for the current goal is (Hfung0 x HxX0).
L181486
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun g0 x) Hg0I0).
L181488
Let k be given.
L181488
Assume HkNat: nat_p k.
L181488
Assume IHk: apply_fun ((State k) 0) x ∈ R.
L181489
We will prove apply_fun ((State (ordsucc k)) 0) x ∈ R.
L181490
We prove the intermediate claim HS: State (ordsucc k) = StepState k (State k).
L181492
An exact proof term for the current goal is (nat_primrec_S BaseState StepState k HkNat).
L181492
rewrite the current goal using HS (from left to right).
L181493
Set st to be the term State k.
L181494
Set g to be the term st 0.
L181495
Set rpack to be the term st 1.
L181496
Set r to be the term rpack 0.
L181497
Set c to be the term rpack 1.
L181498
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181501
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181506
Set cNew to be the term mul_SNo c den.
L181507
We prove the intermediate claim Hdef: StepState k st = (gNew,(rNew,cNew)).
Use reflexivity.
L181509
We prove the intermediate claim HgStep: (StepState k st) 0 = gNew.
L181511
rewrite the current goal using Hdef (from left to right).
L181511
An exact proof term for the current goal is (tuple_2_0_eq gNew (rNew,cNew)).
L181512
rewrite the current goal using HgStep (from left to right).
L181513
We prove the intermediate claim HxX0: x ∈ X.
L181515
An exact proof term for the current goal is (HAsubX x HxA).
L181515
We prove the intermediate claim HgEq: g = (State k) 0.
Use reflexivity.
L181517
We prove the intermediate claim HgxR: apply_fun g x ∈ R.
L181519
rewrite the current goal using HgEq (from left to right).
L181519
An exact proof term for the current goal is IHk.
L181520
We prove the intermediate claim HkO: k ∈ ω.
L181522
An exact proof term for the current goal is (nat_p_omega k HkNat).
L181522
We prove the intermediate claim HrEq: r = ((State k) 1) 0.
Use reflexivity.
L181524
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L181526
rewrite the current goal using HrEq (from left to right).
L181526
An exact proof term for the current goal is (HInv_r_contI k HkO).
L181527
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third).
L181534
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L181534
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L181536
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x0 = minus_SNo one_third)) (∀x0 : set, x0 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x0 = one_third) Hu_pack)).
L181546
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L181548
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L181548
We prove the intermediate claim HuxI0: apply_fun (u_of r) x ∈ I0.
L181550
An exact proof term for the current goal is (Hu_fun x HxX0).
L181550
We prove the intermediate claim HuxR: apply_fun (u_of r) x ∈ R.
L181552
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x) HuxI0).
L181552
We prove the intermediate claim HcEq: c = ((State k) 1) 1.
Use reflexivity.
L181554
We prove the intermediate claim HcR: c ∈ R.
L181556
rewrite the current goal using HcEq (from left to right).
L181556
An exact proof term for the current goal is (andEL (((State k) 1) 1 ∈ R) (0 < ((State k) 1) 1) (HInv_cpos k HkO)).
L181560
We prove the intermediate claim HcorrEq: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = mul_SNo (apply_fun (u_of r) x) c.
L181564
We prove the intermediate claim Hcomp: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x = apply_fun (mul_const_fun c) (apply_fun (u_of r) x).
L181567
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x HxX0).
L181567
rewrite the current goal using Hcomp (from left to right).
L181568
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x) HcR HuxR).
L181569
We prove the intermediate claim HcorrR: apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x ∈ R.
L181571
rewrite the current goal using HcorrEq (from left to right).
L181571
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x) HuxR c HcR).
L181572
We prove the intermediate claim HgNewEval: apply_fun gNew x = add_SNo (apply_fun g x) (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x).
L181576
An exact proof term for the current goal is (add_of_pair_map_apply X g (compose_fun X (u_of r) (mul_const_fun c)) x HxX0 HgxR HcorrR).
L181576
rewrite the current goal using HgNewEval (from left to right).
L181577
An exact proof term for the current goal is (real_add_SNo (apply_fun g x) HgxR (apply_fun (compose_fun X (u_of r) (mul_const_fun c)) x) HcorrR).
L181580
An exact proof term for the current goal is (HgR_nat n (omega_nat_p n HnO)).
L181581
Apply andI to the current goal.
L181583
We will prove (metric_on R R_bounded_metric ∧ sequence_on seq2 R) ∧ lim ∈ R.
L181583
Apply andI to the current goal.
L181585
We will prove metric_on R R_bounded_metric ∧ sequence_on seq2 R.
L181585
Apply andI to the current goal.
L181587
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L181588
An exact proof term for the current goal is Hseq2On.
L181589
We prove the intermediate claim HlimR: lim ∈ R.
L181590
An exact proof term for the current goal is (Hf_R x HxA).
L181590
An exact proof term for the current goal is HlimR.
L181592
Let eps be given.
L181592
Assume Heps: eps ∈ R ∧ Rlt 0 eps.
L181592
We prove the intermediate claim HepsR: eps ∈ R.
(*** ABY: try_aby at line 132949 column 1 failed ***)
(*** TODO: use the Step II residual invariant on A to get abs((State n)0 x - f x) <= ((State n)1)1,
then pick N so that ((State n)1)1 < min(eps,one_third) and conclude using
abs_lt_lt1_imp_R_bounded_distance_lt. ***)
L181598
An exact proof term for the current goal is (andEL (eps ∈ R) (Rlt 0 eps) Heps).
L181598
We prove the intermediate claim HepsPos: Rlt 0 eps.
L181600
An exact proof term for the current goal is (andER (eps ∈ R) (Rlt 0 eps) Heps).
L181600
We prove the intermediate claim HepsS: SNo eps.
L181602
An exact proof term for the current goal is (real_SNo eps HepsR).
L181602
Apply (SNoLt_trichotomy_or_impred eps 1 HepsS SNo_1 (∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps)) to the current goal.
L181607
Assume HepsLt1S: eps < 1.
L181607
We prove the intermediate claim HepsLt1: Rlt eps 1.
(*** TODO: show eps-step for eps<1 using the residual bound on A and a decay bound for ((State n)1)1. ***)
L181610
An exact proof term for the current goal is (RltI eps 1 HepsR real_1 HepsLt1S).
L181610
We prove the intermediate claim Hex_c_small: ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps.
L181615
We prove the intermediate claim HexK: ∃K ∈ ω, eps_ K < eps.
(*** reduce to a dyadic bound eps_ K and a geometric bound for the scalar component ***)
L181617
An exact proof term for the current goal is (exists_eps_lt_pos_Euclid eps HepsR HepsPos).
L181617
Apply HexK to the current goal.
L181618
Let K be given.
L181619
Assume HK.
L181619
We prove the intermediate claim HKomega: K ∈ ω.
L181621
An exact proof term for the current goal is (andEL (K ∈ ω) (eps_ K < eps) HK).
L181621
We prove the intermediate claim HepsKltEpsS: eps_ K < eps.
L181623
An exact proof term for the current goal is (andER (K ∈ ω) (eps_ K < eps) HK).
L181623
We prove the intermediate claim HepsKR: eps_ K ∈ R.
L181625
An exact proof term for the current goal is (SNoS_omega_real (eps_ K) (SNo_eps_SNoS_omega K HKomega)).
L181625
We prove the intermediate claim HepsKltEps: Rlt (eps_ K) eps.
L181627
An exact proof term for the current goal is (RltI (eps_ K) eps HepsKR HepsR HepsKltEpsS).
L181627
We prove the intermediate claim HexN: ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K.
(*** remaining goal: find N with ((State n)1)1 < eps_ K uniformly for n>=N ***)
L181632
We prove the intermediate claim HKnat: nat_p K.
L181633
An exact proof term for the current goal is (omega_nat_p K HKomega).
L181633
We prove the intermediate claim HKcase: K = 0 ∨ ∃k : set, nat_p k ∧ K = ordsucc k.
L181635
An exact proof term for the current goal is (nat_inv K HKnat).
L181635
Apply HKcase to the current goal.
L181637
Assume HK0: K = 0.
L181637
We use 0 to witness the existential quantifier.
L181638
Apply andI to the current goal.
L181640
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L181641
Let n be given.
L181641
Assume HnO: n ∈ ω.
L181641
Assume H0sub: 0 ⊆ n.
L181642
We will prove ((State n) 1) 1 < eps_ K.
L181643
rewrite the current goal using HK0 (from left to right).
L181644
rewrite the current goal using eps_0_1 (from left to right).
L181645
An exact proof term for the current goal is (HInv_c_lt1 n HnO).
L181647
Assume Hexk: ∃k : set, nat_p k ∧ K = ordsucc k.
L181647
Apply Hexk to the current goal.
L181648
Let k be given.
L181649
Assume Hkconj: nat_p k ∧ K = ordsucc k.
L181649
We prove the intermediate claim HkNat: nat_p k.
L181651
An exact proof term for the current goal is (andEL (nat_p k) (K = ordsucc k) Hkconj).
L181651
We prove the intermediate claim HKeq: K = ordsucc k.
L181653
An exact proof term for the current goal is (andER (nat_p k) (K = ordsucc k) Hkconj).
L181653
Set N0 to be the term add_nat K K.
L181655
We use N0 to witness the existential quantifier.
(*** TODO: prove geometric decay of the scalar component and compare with eps_(ordsucc k) ***)
L181656
Apply andI to the current goal.
L181658
We prove the intermediate claim HN0O: N0 ∈ ω.
L181659
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L181659
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L181660
An exact proof term for the current goal is HN0O.
L181662
Let n be given.
L181662
Assume HnO: n ∈ ω.
L181662
Assume HN0sub: N0 ⊆ n.
L181663
We prove the intermediate claim HKnat: nat_p K.
(*** show ((State n) 1) 1 < eps_ K from a dyadic bound at N0 and monotone decay ***)
L181666
An exact proof term for the current goal is (omega_nat_p K HKomega).
L181666
We prove the intermediate claim HdenS: SNo den.
L181668
An exact proof term for the current goal is (real_SNo den HdenR).
L181668
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L181670
We prove the intermediate claim HdenLt1: den < 1.
L181672
rewrite the current goal using HdenDef (from left to right).
L181672
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L181673
Set den2 to be the term mul_SNo den den.
L181674
We prove the intermediate claim Hden2S: SNo den2.
L181676
An exact proof term for the current goal is (SNo_mul_SNo den den HdenS HdenS).
L181676
We prove the intermediate claim Hden2Lt_eps1: den2 < eps_ 1.
L181678
rewrite the current goal using HdenDef (from left to right) at position 1.
L181678
rewrite the current goal using HdenDef (from left to right) at position 2.
L181679
An exact proof term for the current goal is two_thirds_sq_lt_eps_1.
L181680
We prove the intermediate claim Hc_step: ∀m : set, nat_p m → ((State (ordsucc m)) 1) 1 = mul_SNo (((State m) 1) 1) den.
(*** scalar recursion: c_{S m} = c_m times den ***)
L181684
Let m be given.
L181684
Assume HmNat: nat_p m.
L181684
We prove the intermediate claim HS: State (ordsucc m) = StepState m (State m).
L181686
An exact proof term for the current goal is (nat_primrec_S BaseState StepState m HmNat).
L181686
rewrite the current goal using HS (from left to right).
L181687
Set st to be the term State m.
L181688
Set g to be the term st 0.
L181689
Set rpack to be the term st 1.
L181690
Set r to be the term rpack 0.
L181691
Set c to be the term rpack 1.
L181692
Set gNew to be the term compose_fun X (pair_map X g (compose_fun X (u_of r) (mul_const_fun c))) add_fun_R.
L181695
Set rNew to be the term compose_fun A (compose_fun A (pair_map A r (compose_fun A (u_of r) neg_fun)) add_fun_R) (div_const_fun den).
L181700
Set cNew to be the term mul_SNo c den.
L181701
We prove the intermediate claim Hdef: StepState m st = (gNew,(rNew,cNew)).
Use reflexivity.
L181703
We prove the intermediate claim HtEq: ((StepState m st) 1) 1 = cNew.
L181705
We prove the intermediate claim Hinner: (StepState m st) 1 = (rNew,cNew).
L181706
rewrite the current goal using Hdef (from left to right).
L181706
An exact proof term for the current goal is (tuple_2_1_eq gNew (rNew,cNew)).
L181707
rewrite the current goal using Hinner (from left to right).
L181708
An exact proof term for the current goal is (tuple_2_1_eq rNew cNew).
L181709
rewrite the current goal using HtEq (from left to right).
L181710
We prove the intermediate claim HcEq: c = ((State m) 1) 1.
Use reflexivity.
L181712
rewrite the current goal using HcEq (from left to right).
Use reflexivity.
L181714
We prove the intermediate claim Hc_succ_lt: ∀t : set, t ∈ ω → ((State (ordsucc t)) 1) 1 < ((State t) 1) 1.
(*** monotone decay: c_{S t + a} < c_{t + a} for t in omega ***)
L181718
Let t be given.
L181718
Assume HtO: t ∈ ω.
L181718
We prove the intermediate claim HtNat: nat_p t.
L181720
An exact proof term for the current goal is (omega_nat_p t HtO).
L181720
We prove the intermediate claim HcEq: ((State (ordsucc t)) 1) 1 = mul_SNo (((State t) 1) 1) den.
L181722
An exact proof term for the current goal is (Hc_step t HtNat).
L181722
rewrite the current goal using HcEq (from left to right).
L181723
We prove the intermediate claim HctR: ((State t) 1) 1 ∈ R.
L181725
An exact proof term for the current goal is (andEL (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L181725
We prove the intermediate claim HctS: SNo (((State t) 1) 1).
L181727
An exact proof term for the current goal is (real_SNo (((State t) 1) 1) HctR).
L181727
We prove the intermediate claim HctPos: 0 < (((State t) 1) 1).
L181729
An exact proof term for the current goal is (andER (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L181729
We prove the intermediate claim HmulLt: mul_SNo den (((State t) 1) 1) < ((State t) 1) 1.
L181731
An exact proof term for the current goal is (mul_SNo_Lt1_pos_Lt den (((State t) 1) 1) HdenS HctS HdenLt1 HctPos).
L181731
We prove the intermediate claim HmulEq: mul_SNo den (((State t) 1) 1) = mul_SNo (((State t) 1) 1) den.
L181733
An exact proof term for the current goal is (mul_SNo_com den (((State t) 1) 1) HdenS HctS).
L181733
rewrite the current goal using HmulEq (from right to left).
L181734
An exact proof term for the current goal is HmulLt.
L181735
We prove the intermediate claim Hc_even_lt: ∀t : set, nat_p t → ((State (add_nat t t)) 1) 1 < eps_ t.
(*** dyadic bound at even indices: c_{K+K} < eps_K ***)
L181739
Apply nat_ind to the current goal.
L181740
We will prove ((State (add_nat 0 0)) 1) 1 < eps_ 0.
L181740
rewrite the current goal using (add_nat_0R 0) (from left to right) at position 1.
L181741
rewrite the current goal using eps_0_1 (from left to right).
L181742
An exact proof term for the current goal is (HInv_c_lt1 0 (nat_p_omega 0 nat_0)).
L181744
Let t be given.
L181744
Assume HtNat: nat_p t.
L181744
Assume IH: ((State (add_nat t t)) 1) 1 < eps_ t.
L181745
We will prove ((State (add_nat (ordsucc t) (ordsucc t))) 1) 1 < eps_ (ordsucc t).
L181746
Set N to be the term add_nat t t.
L181747
We prove the intermediate claim HNnat: nat_p N.
L181749
An exact proof term for the current goal is (add_nat_p t HtNat t HtNat).
L181749
We prove the intermediate claim Hidx: add_nat (ordsucc t) (ordsucc t) = ordsucc (ordsucc N).
L181751
rewrite the current goal using (add_nat_SL t HtNat (ordsucc t) (nat_ordsucc t HtNat)) (from left to right).
L181751
rewrite the current goal using (add_nat_SR t t HtNat) (from left to right).
Use reflexivity.
L181753
rewrite the current goal using Hidx (from left to right).
L181754
We prove the intermediate claim HN0: N ∈ ω.
L181756
An exact proof term for the current goal is (nat_p_omega N HNnat).
L181756
We prove the intermediate claim HcSN: ((State (ordsucc N)) 1) 1 = mul_SNo (((State N) 1) 1) den.
L181758
An exact proof term for the current goal is (Hc_step N HNnat).
L181758
We prove the intermediate claim HcSSN: ((State (ordsucc (ordsucc N))) 1) 1 = mul_SNo (mul_SNo (((State N) 1) 1) den) den.
L181761
We prove the intermediate claim HSNnat: nat_p (ordsucc N).
L181762
An exact proof term for the current goal is (nat_ordsucc N HNnat).
L181762
rewrite the current goal using (Hc_step (ordsucc N) HSNnat) (from left to right).
L181763
rewrite the current goal using HcSN (from left to right).
Use reflexivity.
L181765
rewrite the current goal using HcSSN (from left to right).
L181766
rewrite the current goal using (mul_SNo_assoc (((State N) 1) 1) den den (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))) HdenS HdenS) (from right to left).
L181769
We prove the intermediate claim Hden2Def: den2 = mul_SNo den den.
Use reflexivity.
L181771
rewrite the current goal using Hden2Def (from right to left).
L181772
We prove the intermediate claim HcNpos: 0 < ((State N) 1) 1.
L181774
An exact proof term for the current goal is (andER (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0)).
L181774
We prove the intermediate claim HcNS: SNo (((State N) 1) 1).
L181776
An exact proof term for the current goal is (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))).
L181776
rewrite the current goal using (mul_SNo_com (((State N) 1) 1) den2 HcNS Hden2S) (from left to right).
L181777
We prove the intermediate claim Heps1R: (eps_ 1) ∈ R.
L181779
An exact proof term for the current goal is (SNoS_omega_real (eps_ 1) (SNo_eps_SNoS_omega 1 (nat_p_omega 1 nat_1))).
L181779
We prove the intermediate claim Heps1S: SNo (eps_ 1).
L181781
An exact proof term for the current goal is (real_SNo (eps_ 1) Heps1R).
L181781
We prove the intermediate claim Heps1pos: 0 < eps_ 1.
L181783
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L181783
We prove the intermediate claim Hmul1: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (((State N) 1) 1).
L181785
An exact proof term for the current goal is (pos_mul_SNo_Lt' den2 (eps_ 1) (((State N) 1) 1) Hden2S Heps1S HcNS HcNpos Hden2Lt_eps1).
L181786
We prove the intermediate claim Heps_tR: (eps_ t) ∈ R.
L181788
An exact proof term for the current goal is (SNoS_omega_real (eps_ t) (SNo_eps_SNoS_omega t (nat_p_omega t HtNat))).
L181788
We prove the intermediate claim Heps_tS: SNo (eps_ t).
L181790
An exact proof term for the current goal is (real_SNo (eps_ t) Heps_tR).
L181790
We prove the intermediate claim Hmul2: mul_SNo (eps_ 1) (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L181792
We prove the intermediate claim HcomL: mul_SNo (eps_ 1) (((State N) 1) 1) = mul_SNo (((State N) 1) 1) (eps_ 1).
L181793
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (((State N) 1) 1) Heps1S HcNS).
L181793
We prove the intermediate claim HcomR: mul_SNo (eps_ 1) (eps_ t) = mul_SNo (eps_ t) (eps_ 1).
L181795
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (eps_ t) Heps1S Heps_tS).
L181795
rewrite the current goal using HcomL (from left to right).
L181796
rewrite the current goal using HcomR (from left to right).
L181797
An exact proof term for the current goal is (pos_mul_SNo_Lt' (((State N) 1) 1) (eps_ t) (eps_ 1) HcNS Heps_tS Heps1S Heps1pos IH).
L181799
We prove the intermediate claim HmulTra: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L181801
An exact proof term for the current goal is (SNoLt_tra (mul_SNo den2 (((State N) 1) 1)) (mul_SNo (eps_ 1) (((State N) 1) 1)) (mul_SNo (eps_ 1) (eps_ t)) (SNo_mul_SNo den2 (((State N) 1) 1) Hden2S HcNS) (SNo_mul_SNo (eps_ 1) (((State N) 1) 1) Heps1S HcNS) (SNo_mul_SNo (eps_ 1) (eps_ t) Heps1S Heps_tS) Hmul1 Hmul2).
L181807
We prove the intermediate claim HepsEq: mul_SNo (eps_ 1) (eps_ t) = eps_ (ordsucc t).
L181809
rewrite the current goal using (mul_SNo_eps_eps_add_SNo 1 (nat_p_omega 1 nat_1) t (nat_p_omega t HtNat)) (from left to right).
L181809
We prove the intermediate claim Hordt: ordinal t.
L181811
An exact proof term for the current goal is (nat_p_ordinal t HtNat).
L181811
rewrite the current goal using (ordinal_ordsucc_SNo_eq t Hordt) (from right to left).
Use reflexivity.
L181813
rewrite the current goal using HepsEq (from right to left).
L181814
An exact proof term for the current goal is HmulTra.
L181815
We prove the intermediate claim HcN0: ((State N0) 1) 1 < eps_ K.
L181817
An exact proof term for the current goal is (Hc_even_lt K HKnat).
L181817
We prove the intermediate claim HnNat: nat_p n.
(*** show n is N0 plus a natural shift and reduce to k=0 or successor ***)
L181820
An exact proof term for the current goal is (omega_nat_p n HnO).
L181820
We prove the intermediate claim HN0O: N0 ∈ ω.
L181822
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L181822
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L181823
We prove the intermediate claim HN0Nat: nat_p N0.
L181825
An exact proof term for the current goal is (omega_nat_p N0 HN0O).
L181825
We prove the intermediate claim Hexk: ∃k : set, nat_p k ∧ n = add_nat k N0.
L181827
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat n HnNat HN0sub).
L181827
Apply Hexk to the current goal.
L181828
Let k1 be given.
L181829
Assume Hk1.
L181829
We prove the intermediate claim Hk1Nat: nat_p k1.
L181831
An exact proof term for the current goal is (andEL (nat_p k1) (n = add_nat k1 N0) Hk1).
L181831
We prove the intermediate claim HnEq: n = add_nat k1 N0.
L181833
An exact proof term for the current goal is (andER (nat_p k1) (n = add_nat k1 N0) Hk1).
L181833
rewrite the current goal using HnEq (from left to right).
L181834
We prove the intermediate claim Hkcase: k1 = 0 ∨ ∃j : set, nat_p j ∧ k1 = ordsucc j.
L181836
An exact proof term for the current goal is (nat_inv k1 Hk1Nat).
L181836
Apply Hkcase to the current goal.
L181838
Assume Hk10: k1 = 0.
L181838
rewrite the current goal using Hk10 (from left to right).
L181839
rewrite the current goal using (add_nat_0L N0 HN0Nat) (from left to right).
L181840
An exact proof term for the current goal is HcN0.
L181842
Assume Hk1S: ∃j : set, nat_p j ∧ k1 = ordsucc j.
L181842
Apply Hk1S to the current goal.
L181843
Let j be given.
L181844
Assume Hj.
L181844
We prove the intermediate claim HjNat: nat_p j.
L181846
An exact proof term for the current goal is (andEL (nat_p j) (k1 = ordsucc j) Hj).
L181846
We prove the intermediate claim Hk1Eq: k1 = ordsucc j.
L181848
An exact proof term for the current goal is (andER (nat_p j) (k1 = ordsucc j) Hj).
L181848
rewrite the current goal using Hk1Eq (from left to right).
L181849
We prove the intermediate claim Hdec: ∀j : set, nat_p j → ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L181852
Apply nat_ind to the current goal.
L181853
We will prove ((State (add_nat (ordsucc 0) N0)) 1) 1 < ((State N0) 1) 1.
L181853
rewrite the current goal using (add_nat_SL 0 nat_0 N0 HN0Nat) (from left to right).
L181854
rewrite the current goal using (add_nat_0L N0 HN0Nat) (from left to right).
L181855
An exact proof term for the current goal is (Hc_succ_lt N0 HN0O).
L181857
Let j be given.
L181857
Assume HjNat: nat_p j.
L181857
Assume IHj: ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L181858
We will prove ((State (add_nat (ordsucc (ordsucc j)) N0)) 1) 1 < ((State N0) 1) 1.
L181859
rewrite the current goal using (add_nat_SL (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat) (from left to right).
L181860
We prove the intermediate claim HstepLt: ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 < ((State (add_nat (ordsucc j) N0)) 1) 1.
L181862
An exact proof term for the current goal is (Hc_succ_lt (add_nat (ordsucc j) N0) (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat))).
L181864
We prove the intermediate claim HaO: ordsucc (add_nat (ordsucc j) N0) ∈ ω.
L181866
An exact proof term for the current goal is (nat_p_omega (ordsucc (add_nat (ordsucc j) N0)) (nat_ordsucc (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat))).
L181868
We prove the intermediate claim HbO: add_nat (ordsucc j) N0 ∈ ω.
L181870
An exact proof term for the current goal is (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat)).
L181871
We prove the intermediate claim HaR: ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 ∈ R.
L181873
An exact proof term for the current goal is (andEL (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1 ∈ R) (0 < ((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) (HInv_cpos (ordsucc (add_nat (ordsucc j) N0)) HaO)).
L181875
We prove the intermediate claim HbR: ((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R.
L181877
An exact proof term for the current goal is (andEL (((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R) (0 < ((State (add_nat (ordsucc j) N0)) 1) 1) (HInv_cpos (add_nat (ordsucc j) N0) HbO)).
L181879
We prove the intermediate claim HcR0: ((State N0) 1) 1 ∈ R.
L181881
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L181881
We prove the intermediate claim HaS: SNo (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1).
L181883
An exact proof term for the current goal is (real_SNo (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) HaR).
L181883
We prove the intermediate claim HbS: SNo (((State (add_nat (ordsucc j) N0)) 1) 1).
L181885
An exact proof term for the current goal is (real_SNo (((State (add_nat (ordsucc j) N0)) 1) 1) HbR).
L181885
We prove the intermediate claim HcS0: SNo (((State N0) 1) 1).
L181887
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcR0).
L181887
An exact proof term for the current goal is (SNoLt_tra (((State (ordsucc (add_nat (ordsucc j) N0))) 1) 1) (((State (add_nat (ordsucc j) N0)) 1) 1) (((State N0) 1) 1) HaS HbS HcS0 HstepLt IHj).
L181891
We prove the intermediate claim HltN0: ((State (add_nat (ordsucc j) N0)) 1) 1 < ((State N0) 1) 1.
L181893
An exact proof term for the current goal is (Hdec j HjNat).
L181893
We prove the intermediate claim HxO: add_nat (ordsucc j) N0 ∈ ω.
L181895
An exact proof term for the current goal is (nat_p_omega (add_nat (ordsucc j) N0) (add_nat_p (ordsucc j) (nat_ordsucc j HjNat) N0 HN0Nat)).
L181896
We prove the intermediate claim HxR: ((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R.
L181898
An exact proof term for the current goal is (andEL (((State (add_nat (ordsucc j) N0)) 1) 1 ∈ R) (0 < ((State (add_nat (ordsucc j) N0)) 1) 1) (HInv_cpos (add_nat (ordsucc j) N0) HxO)).
L181900
We prove the intermediate claim HxS: SNo (((State (add_nat (ordsucc j) N0)) 1) 1).
L181902
An exact proof term for the current goal is (real_SNo (((State (add_nat (ordsucc j) N0)) 1) 1) HxR).
L181902
We prove the intermediate claim HN0R: ((State N0) 1) 1 ∈ R.
L181904
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L181904
We prove the intermediate claim HN0S: SNo (((State N0) 1) 1).
L181906
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HN0R).
L181906
We prove the intermediate claim HepsKS: SNo (eps_ K).
L181908
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L181908
An exact proof term for the current goal is (SNoLt_tra (((State (add_nat (ordsucc j) N0)) 1) 1) (((State N0) 1) 1) (eps_ K) HxS HN0S HepsKS HltN0 HcN0).
L181912
Apply HexN to the current goal.
L181913
Let N be given.
L181914
Assume HN.
L181914
We use N to witness the existential quantifier.
L181915
Apply andI to the current goal.
L181917
An exact proof term for the current goal is (andEL (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K) HN).
L181920
Let n be given.
L181920
Assume HnO: n ∈ ω.
L181920
Assume HNsub: N ⊆ n.
L181921
We prove the intermediate claim HNprop: ∀t : set, t ∈ ω → N ⊆ t → ((State t) 1) 1 < eps_ K.
L181924
An exact proof term for the current goal is (andER (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps_ K) HN).
L181926
We prove the intermediate claim HcLtEpsK: ((State n) 1) 1 < eps_ K.
L181928
An exact proof term for the current goal is (HNprop n HnO HNsub).
L181928
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L181930
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L181930
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L181932
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L181932
We prove the intermediate claim HepsKS: SNo (eps_ K).
L181934
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L181934
We prove the intermediate claim HepsS0: SNo eps.
L181936
An exact proof term for the current goal is (real_SNo eps HepsR).
L181936
An exact proof term for the current goal is (SNoLt_tra (((State n) 1) 1) (eps_ K) eps HcS HepsKS HepsS0 HcLtEpsK HepsKltEpsS).
L181938
Apply Hex_c_small to the current goal.
L181939
Let N be given.
L181940
Assume HNpair.
L181940
We use N to witness the existential quantifier.
L181941
Apply andI to the current goal.
L181943
An exact proof term for the current goal is (andEL (N ∈ ω) (∀n : set, n ∈ ω → N ⊆ n → ((State n) 1) 1 < eps) HNpair).
L181946
Let n be given.
L181946
Assume HnO: n ∈ ω.
L181946
Assume HNsub: N ⊆ n.
L181947
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L181948
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L181950
An exact proof term for the current goal is (Hseq2On n HnO).
L181950
We prove the intermediate claim HlimR: lim ∈ R.
L181952
An exact proof term for the current goal is (Hf_R x HxA).
L181952
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L181954
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L181954
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L181957
rewrite the current goal using HdefM (from left to right).
L181958
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L181960
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L181961
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L181962
Set a to be the term apply_fun seq2 n.
L181963
Set b to be the term lim.
L181964
Set t to be the term add_SNo a (minus_SNo b).
L181965
We prove the intermediate claim HmbR: minus_SNo b ∈ R.
L181967
An exact proof term for the current goal is (real_minus_SNo b HlimR).
L181967
We prove the intermediate claim HtR: t ∈ R.
L181969
An exact proof term for the current goal is (real_add_SNo a Hseq2nR (minus_SNo b) HmbR).
L181969
We prove the intermediate claim HtS: SNo t.
L181971
An exact proof term for the current goal is (real_SNo t HtR).
L181971
We prove the intermediate claim HaS: SNo a.
L181973
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L181973
We prove the intermediate claim HbS: SNo b.
L181975
An exact proof term for the current goal is (real_SNo b HlimR).
L181975
We prove the intermediate claim HabsEq: abs_SNo t = abs_SNo (add_SNo b (minus_SNo a)).
L181977
An exact proof term for the current goal is (abs_SNo_dist_swap a b HaS HbS).
L181977
We prove the intermediate claim HaEq: a = apply_fun ((State n) 0) x.
L181979
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L181980
rewrite the current goal using Hseq2def (from left to right).
L181981
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
Use reflexivity.
L181983
We prove the intermediate claim Hident: add_SNo a (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = b.
L181987
rewrite the current goal using HaEq (from left to right).
L181987
An exact proof term for the current goal is (HInv_residual_identity_A n HnO x HxA).
L181988
We prove the intermediate claim HrcEq: add_SNo b (minus_SNo a) = mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L181992
rewrite the current goal using Hident (from right to left) at position 1.
L181992
Set rc to be the term mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L181993
We prove the intermediate claim HrcR: rc ∈ R.
L181995
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L181996
An exact proof term for the current goal is (HInv_r_contI n HnO).
L181996
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L181998
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L181998
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L182000
An exact proof term for the current goal is (Hrfun_on x HxA).
L182000
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L182002
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L182002
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L182004
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182004
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L182005
We prove the intermediate claim HrcS: SNo rc.
L182007
An exact proof term for the current goal is (real_SNo rc HrcR).
L182007
We prove the intermediate claim HaS0: SNo a.
L182009
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L182009
rewrite the current goal using (add_SNo_com a rc HaS0 HrcS) (from left to right) at position 1.
L182010
An exact proof term for the current goal is (add_SNo_minus_R2 rc a HrcS HaS0).
L182011
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L182013
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182013
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L182015
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L182015
We prove the intermediate claim HcLtEps: ((State n) 1) 1 < eps.
L182017
An exact proof term for the current goal is (andER (N ∈ ω) (∀n0 : set, n0 ∈ ω → N ⊆ n0 → ((State n0) 1) 1 < eps) HNpair n HnO HNsub).
L182019
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L182021
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182021
We prove the intermediate claim H0le_c: 0 ≤ ((State n) 1) 1.
L182023
An exact proof term for the current goal is (SNoLtLe 0 (((State n) 1) 1) Hcpos).
L182023
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L182025
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182025
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L182027
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L182027
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L182029
An exact proof term for the current goal is (Hrfun_on x HxA).
L182029
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L182031
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L182031
We prove the intermediate claim HprodI: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ closed_interval (minus_SNo (((State n) 1) 1)) (((State n) 1) 1).
L182034
An exact proof term for the current goal is (mul_nonneg_closed_interval_minus1_1 (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) HrxI HcR H0le_c).
L182035
We prove the intermediate claim Hmcr: (minus_SNo (((State n) 1) 1)) ∈ R.
L182037
An exact proof term for the current goal is (real_minus_SNo (((State n) 1) 1) HcR).
L182037
We prove the intermediate claim HprodR: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ R.
L182039
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L182039
We prove the intermediate claim HprodS: SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L182041
An exact proof term for the current goal is (real_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L182041
We prove the intermediate claim Hbounds: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∧ Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L182045
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo (((State n) 1) 1)) (((State n) 1) 1) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) Hmcr HcR HprodI).
L182046
We prove the intermediate claim HloRle: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L182049
An exact proof term for the current goal is (andEL (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L182052
We prove the intermediate claim HhiRle: Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L182055
An exact proof term for the current goal is (andER (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L182058
We prove the intermediate claim HloLe: minus_SNo (((State n) 1) 1) ≤ mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L182060
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HloRle).
L182062
We prove the intermediate claim HhiLe: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ≤ ((State n) 1) 1.
L182064
An exact proof term for the current goal is (SNoLe_of_Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HhiRle).
L182067
We prove the intermediate claim HabsLe: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ≤ ((State n) 1) 1.
L182069
An exact proof term for the current goal is (abs_SNo_Le_of_bounds (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HprodS HcS HloLe HhiLe).
L182075
We prove the intermediate claim HabsR: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∈ R.
L182077
An exact proof term for the current goal is (abs_SNo_in_R (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L182077
We prove the intermediate claim HabsS: SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))).
L182079
An exact proof term for the current goal is (real_SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) HabsR).
L182079
We prove the intermediate claim HabsLtEps: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) < eps.
L182082
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (((State n) 1) 1) eps HabsS HcS HepsS HabsLe HcLtEps).
L182090
We prove the intermediate claim HabstLtEps: abs_SNo t < eps.
L182092
rewrite the current goal using HabsEq (from left to right).
L182092
rewrite the current goal using HrcEq (from left to right).
L182093
An exact proof term for the current goal is HabsLtEps.
L182094
An exact proof term for the current goal is (abs_lt_lt1_imp_R_bounded_distance_lt a b eps Hseq2nR HlimR HepsR HepsLt1 HabstLtEps).
L182097
Assume HepsEq1: eps = 1.
L182097
rewrite the current goal using HepsEq1 (from left to right).
L182099
We use 0 to witness the existential quantifier.
(*** TODO: show eventually R_bounded_distance < 1 using abs difference < 1 from the residual bound on A. ***)
L182100
Apply andI to the current goal.
L182102
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L182103
Let n be given.
L182103
Assume HnO: n ∈ ω.
L182103
Assume H0sub: 0 ⊆ n.
L182104
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) 1.
L182105
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L182107
An exact proof term for the current goal is (Hseq2On n HnO).
L182107
We prove the intermediate claim HlimR: lim ∈ R.
L182109
An exact proof term for the current goal is (Hf_R x HxA).
L182109
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L182111
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L182111
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L182114
rewrite the current goal using HdefM (from left to right).
L182115
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L182117
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L182118
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L182119
Set a to be the term apply_fun seq2 n.
L182120
Set b to be the term lim.
L182121
Set t to be the term add_SNo a (minus_SNo b).
L182122
We prove the intermediate claim HmbR: minus_SNo b ∈ R.
L182124
An exact proof term for the current goal is (real_minus_SNo b HlimR).
L182124
We prove the intermediate claim HtR: t ∈ R.
L182126
An exact proof term for the current goal is (real_add_SNo a Hseq2nR (minus_SNo b) HmbR).
L182126
We prove the intermediate claim HtS: SNo t.
L182128
An exact proof term for the current goal is (real_SNo t HtR).
L182128
We prove the intermediate claim HaS: SNo a.
L182130
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L182130
We prove the intermediate claim HbS: SNo b.
L182132
An exact proof term for the current goal is (real_SNo b HlimR).
L182132
We prove the intermediate claim HabsEq: abs_SNo t = abs_SNo (add_SNo b (minus_SNo a)).
L182134
An exact proof term for the current goal is (abs_SNo_dist_swap a b HaS HbS).
L182134
We prove the intermediate claim HaEq: a = apply_fun ((State n) 0) x.
L182136
We prove the intermediate claim Hseq2def: seq2 = graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x).
Use reflexivity.
L182137
rewrite the current goal using Hseq2def (from left to right).
L182138
rewrite the current goal using (apply_fun_graph ω (λn0 : set ⇒ apply_fun ((State n0) 0) x) n HnO) (from left to right).
Use reflexivity.
L182140
We prove the intermediate claim Hident: add_SNo a (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) = b.
L182144
rewrite the current goal using HaEq (from left to right).
L182144
An exact proof term for the current goal is (HInv_residual_identity_A n HnO x HxA).
L182145
We prove the intermediate claim HrcEq: add_SNo b (minus_SNo a) = mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L182149
rewrite the current goal using Hident (from right to left) at position 1.
L182149
Set rc to be the term mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L182150
We prove the intermediate claim HrcR: rc ∈ R.
L182152
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L182153
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182153
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L182155
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L182155
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L182157
An exact proof term for the current goal is (Hrfun_on x HxA).
L182157
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L182159
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L182159
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L182161
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182161
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L182162
We prove the intermediate claim HrcS: SNo rc.
L182164
An exact proof term for the current goal is (real_SNo rc HrcR).
L182164
We prove the intermediate claim HaS0: SNo a.
L182166
An exact proof term for the current goal is (real_SNo a Hseq2nR).
L182166
rewrite the current goal using (add_SNo_com a rc HaS0 HrcS) (from left to right) at position 1.
L182167
An exact proof term for the current goal is (add_SNo_minus_R2 rc a HrcS HaS0).
L182168
We prove the intermediate claim HcR: ((State n) 1) 1 ∈ R.
L182170
An exact proof term for the current goal is (andEL (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182170
We prove the intermediate claim HcS: SNo (((State n) 1) 1).
L182172
An exact proof term for the current goal is (real_SNo (((State n) 1) 1) HcR).
L182172
We prove the intermediate claim HcLt1: ((State n) 1) 1 < 1.
L182174
An exact proof term for the current goal is (HInv_c_lt1 n HnO).
L182174
We prove the intermediate claim Hcpos: 0 < ((State n) 1) 1.
L182176
An exact proof term for the current goal is (andER (((State n) 1) 1 ∈ R) (0 < ((State n) 1) 1) (HInv_cpos n HnO)).
L182176
We prove the intermediate claim H0le_c: 0 ≤ ((State n) 1) 1.
L182178
An exact proof term for the current goal is (SNoLtLe 0 (((State n) 1) 1) Hcpos).
L182178
We prove the intermediate claim Hrfun: continuous_map A Ta I Ti (((State n) 1) 0).
L182180
An exact proof term for the current goal is (HInv_r_contI n HnO).
L182180
We prove the intermediate claim Hrfun_on: function_on (((State n) 1) 0) A I.
L182182
An exact proof term for the current goal is (continuous_map_function_on A Ta I Ti (((State n) 1) 0) Hrfun).
L182182
We prove the intermediate claim HrxI: apply_fun (((State n) 1) 0) x ∈ I.
L182184
An exact proof term for the current goal is (Hrfun_on x HxA).
L182184
We prove the intermediate claim HrxR: apply_fun (((State n) 1) 0) x ∈ R.
L182186
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (((State n) 1) 0) x) HrxI).
L182186
We prove the intermediate claim HrxS: SNo (apply_fun (((State n) 1) 0) x).
L182188
An exact proof term for the current goal is (real_SNo (apply_fun (((State n) 1) 0) x) HrxR).
L182188
We prove the intermediate claim HprodI: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ closed_interval (minus_SNo (((State n) 1) 1)) (((State n) 1) 1).
L182191
An exact proof term for the current goal is (mul_nonneg_closed_interval_minus1_1 (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) HrxI HcR H0le_c).
L182192
We prove the intermediate claim Hmcr: (minus_SNo (((State n) 1) 1)) ∈ R.
L182194
An exact proof term for the current goal is (real_minus_SNo (((State n) 1) 1) HcR).
L182194
We prove the intermediate claim HprodR: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ∈ R.
L182196
An exact proof term for the current goal is (real_mul_SNo (apply_fun (((State n) 1) 0) x) HrxR (((State n) 1) 1) HcR).
L182196
We prove the intermediate claim HprodS: SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L182198
An exact proof term for the current goal is (real_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L182198
We prove the intermediate claim Hbounds: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∧ Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L182202
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo (((State n) 1) 1)) (((State n) 1) 1) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) Hmcr HcR HprodI).
L182203
We prove the intermediate claim HloRle: Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)).
L182206
An exact proof term for the current goal is (andEL (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L182209
We prove the intermediate claim HhiRle: Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1).
L182212
An exact proof term for the current goal is (andER (Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1)) Hbounds).
L182215
We prove the intermediate claim HloLe: minus_SNo (((State n) 1) 1) ≤ mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1).
L182217
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo (((State n) 1) 1)) (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HloRle).
L182219
We prove the intermediate claim HhiLe: mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1) ≤ ((State n) 1) 1.
L182221
An exact proof term for the current goal is (SNoLe_of_Rle (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HhiRle).
L182224
We prove the intermediate claim HabsLe: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ≤ ((State n) 1) 1.
L182226
An exact proof term for the current goal is (abs_SNo_Le_of_bounds (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) (((State n) 1) 1) HprodS HcS HloLe HhiLe).
L182232
We prove the intermediate claim HabsR: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) ∈ R.
L182234
An exact proof term for the current goal is (abs_SNo_in_R (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) HprodR).
L182234
We prove the intermediate claim HabsS: SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))).
L182236
An exact proof term for the current goal is (real_SNo (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) HabsR).
L182236
We prove the intermediate claim HabsLt1: abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1)) < 1.
L182239
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) (((State n) 1) 1) 1 HabsS HcS SNo_1 HabsLe HcLt1).
L182247
We prove the intermediate claim HabsLt1R: Rlt (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) 1.
L182249
An exact proof term for the current goal is (RltI (abs_SNo (mul_SNo (apply_fun (((State n) 1) 0) x) (((State n) 1) 1))) 1 HabsR real_1 HabsLt1).
L182254
We prove the intermediate claim HabstLt1: Rlt (abs_SNo t) 1.
L182256
rewrite the current goal using HabsEq (from left to right).
L182256
rewrite the current goal using HrcEq (from left to right).
L182257
An exact proof term for the current goal is HabsLt1R.
L182258
We prove the intermediate claim Hbddef: R_bounded_distance a b = If_i (Rlt (abs_SNo t) 1) (abs_SNo t) 1.
Use reflexivity.
L182261
rewrite the current goal using Hbddef (from left to right).
L182262
rewrite the current goal using (If_i_1 (Rlt (abs_SNo t) 1) (abs_SNo t) 1 HabstLt1) (from left to right).
L182263
An exact proof term for the current goal is HabstLt1.
L182265
Assume H1LtEpsS: 1 < eps.
L182265
We prove the intermediate claim H1LtEps: Rlt 1 eps.
L182267
An exact proof term for the current goal is (RltI 1 eps real_1 HepsR H1LtEpsS).
L182267
We use 0 to witness the existential quantifier.
L182268
Apply andI to the current goal.
L182270
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L182271
Let n be given.
L182271
Assume HnO: n ∈ ω.
L182271
Assume H0sub: 0 ⊆ n.
L182272
We will prove Rlt (apply_fun R_bounded_metric (apply_fun seq2 n,lim)) eps.
L182273
We prove the intermediate claim Hseq2nR: apply_fun seq2 n ∈ R.
L182275
An exact proof term for the current goal is (Hseq2On n HnO).
L182275
We prove the intermediate claim HlimR: lim ∈ R.
L182277
An exact proof term for the current goal is (Hf_R x HxA).
L182277
We prove the intermediate claim Hpair: (apply_fun seq2 n,lim) ∈ setprod R R.
L182279
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun seq2 n) lim Hseq2nR HlimR).
L182279
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L182282
rewrite the current goal using HdefM (from left to right).
L182283
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun seq2 n,lim) Hpair) (from left to right).
L182285
rewrite the current goal using (tuple_2_0_eq (apply_fun seq2 n) lim) (from left to right).
L182286
rewrite the current goal using (tuple_2_1_eq (apply_fun seq2 n) lim) (from left to right).
L182287
We prove the intermediate claim Hle1: Rle (R_bounded_distance (apply_fun seq2 n) lim) 1.
L182289
An exact proof term for the current goal is (R_bounded_distance_le_1 (apply_fun seq2 n) lim Hseq2nR HlimR).
L182289
An exact proof term for the current goal is (Rle_Rlt_tra (R_bounded_distance (apply_fun seq2 n) lim) 1 eps Hle1 H1LtEps).
L182290
An exact proof term for the current goal is (sequence_converges_metric_imp_converges_to_metric_topology R R_bounded_metric seq2 lim HconvM).
L182292
An exact proof term for the current goal is Hseq2_conv.
L182294
Let eps be given.
L182295
Assume HepsR: eps ∈ R.
L182295
Assume HepsPos: Rlt 0 eps.
L182295
We will prove ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps.
(*** uniform Cauchy (to be proved from a geometric tail estimate for the correction terms) ***)
L182299
We prove the intermediate claim HepsS: SNo eps.
(*** TODO: derive from the recursive definition of State and the bounds u_of(r) :e I0 with coefficients (2/3)^k. ***)
(*** TODO: pick N using an eps_N tail bound smaller than eps and compare in the bounded metric. ***)
L182303
An exact proof term for the current goal is (real_SNo eps HepsR).
L182303
We prove the intermediate claim Huc_small: ∀eps0 : set, eps0 ∈ R → Rlt 0 eps0 → Rlt eps0 1 → ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps0.
L182310
Let eps0 be given.
L182310
Assume Heps0R: eps0 ∈ R.
L182310
Assume Heps0Pos: Rlt 0 eps0.
L182310
Assume Heps0Lt1: Rlt eps0 1.
L182310
We prove the intermediate claim Heps0S: SNo eps0.
L182312
An exact proof term for the current goal is (real_SNo eps0 Heps0R).
L182312
Set eta to be the term mul_SNo eps0 (eps_ 1).
L182313
We prove the intermediate claim HetaR: eta ∈ R.
L182315
An exact proof term for the current goal is (real_mul_SNo eps0 Heps0R (eps_ 1) eps_1_in_R).
L182315
We prove the intermediate claim HetaS: SNo eta.
L182317
An exact proof term for the current goal is (real_SNo eta HetaR).
L182317
We prove the intermediate claim Heps0PosS: 0 < eps0.
L182319
An exact proof term for the current goal is (RltE_lt 0 eps0 Heps0Pos).
L182319
We prove the intermediate claim Heps1S: SNo (eps_ 1).
L182321
An exact proof term for the current goal is (real_SNo (eps_ 1) eps_1_in_R).
L182321
We prove the intermediate claim Heps1PosS: 0 < eps_ 1.
L182323
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L182323
We prove the intermediate claim HetaPosS: 0 < eta.
L182325
An exact proof term for the current goal is (mul_SNo_pos_pos eps0 (eps_ 1) Heps0S Heps1S Heps0PosS Heps1PosS).
L182325
We prove the intermediate claim HetaPos: Rlt 0 eta.
L182327
An exact proof term for the current goal is (RltI 0 eta real_0 HetaR HetaPosS).
L182327
We prove the intermediate claim HexK: ∃K ∈ ω, eps_ K < eta.
L182329
An exact proof term for the current goal is (exists_eps_lt_pos_Euclid eta HetaR HetaPos).
L182329
Apply HexK to the current goal.
L182330
Let K be given.
L182331
Assume HK.
L182331
We prove the intermediate claim HKomega: K ∈ ω.
L182333
An exact proof term for the current goal is (andEL (K ∈ ω) (eps_ K < eta) HK).
L182333
We prove the intermediate claim HKnat: nat_p K.
L182335
An exact proof term for the current goal is (omega_nat_p K HKomega).
L182335
We prove the intermediate claim HepsKltEtaS: eps_ K < eta.
L182337
An exact proof term for the current goal is (andER (K ∈ ω) (eps_ K < eta) HK).
L182337
We prove the intermediate claim HepsKR: eps_ K ∈ R.
L182339
An exact proof term for the current goal is (SNoS_omega_real (eps_ K) (SNo_eps_SNoS_omega K HKomega)).
L182339
We prove the intermediate claim HepsKS: SNo (eps_ K).
L182341
An exact proof term for the current goal is (real_SNo (eps_ K) HepsKR).
L182341
Set N0 to be the term add_nat K K.
L182342
We prove the intermediate claim HN0O: N0 ∈ ω.
L182344
rewrite the current goal using (add_nat_add_SNo K HKomega K HKomega) (from left to right).
L182344
An exact proof term for the current goal is (add_SNo_In_omega K HKomega K HKomega).
L182345
We prove the intermediate claim HN0Nat: nat_p N0.
L182347
An exact proof term for the current goal is (omega_nat_p N0 HN0O).
L182347
We prove the intermediate claim HdenS: SNo den.
L182349
An exact proof term for the current goal is (real_SNo den HdenR).
L182349
We prove the intermediate claim HdenDef: den = two_thirds.
Use reflexivity.
L182351
We prove the intermediate claim HdenLt1: den < 1.
L182353
rewrite the current goal using HdenDef (from left to right).
L182353
An exact proof term for the current goal is (RltE_lt two_thirds 1 Rlt_two_thirds_1).
L182354
Set den2 to be the term mul_SNo den den.
L182355
We prove the intermediate claim Hden2S: SNo den2.
L182357
An exact proof term for the current goal is (SNo_mul_SNo den den HdenS HdenS).
L182357
We prove the intermediate claim Hden2Lt_eps1: den2 < eps_ 1.
L182359
rewrite the current goal using HdenDef (from left to right) at position 1.
L182359
rewrite the current goal using HdenDef (from left to right) at position 2.
L182360
An exact proof term for the current goal is two_thirds_sq_lt_eps_1.
L182361
We prove the intermediate claim Hc_step: ∀m : set, nat_p m → ((State (ordsucc m)) 1) 1 = mul_SNo (((State m) 1) 1) den.
L182364
Let m be given.
L182364
Assume HmNat: nat_p m.
L182364
We prove the intermediate claim HS: State (ordsucc m) = StepState m (State m).
L182366
An exact proof term for the current goal is (nat_primrec_S BaseState StepState m HmNat).
L182366
rewrite the current goal using HS (from left to right).
L182367
Set st to be the term State m.
L182368
Set c to be the term (st 1) 1.
L182369
Set cNew to be the term mul_SNo c den.
L182370
We prove the intermediate claim Hdef: StepState m st = (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun c))) add_fun_R,(compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew)).
Use reflexivity.
L182383
We prove the intermediate claim HtEq: ((StepState m st) 1) 1 = cNew.
L182385
We prove the intermediate claim Hinner: (StepState m st) 1 = (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew).
L182392
rewrite the current goal using Hdef (from left to right).
L182392
An exact proof term for the current goal is (tuple_2_1_eq (compose_fun X (pair_map X (st 0) (compose_fun X (u_of ((st 1) 0)) (mul_const_fun c))) add_fun_R) (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),cNew)).
L182404
rewrite the current goal using Hinner (from left to right).
L182405
An exact proof term for the current goal is (tuple_2_1_eq (compose_fun A (compose_fun A (pair_map A ((st 1) 0) (compose_fun A (u_of ((st 1) 0)) neg_fun)) add_fun_R) (div_const_fun den)) cNew).
L182412
rewrite the current goal using HtEq (from left to right).
L182413
We prove the intermediate claim HstEq: st = State m.
Use reflexivity.
L182415
We prove the intermediate claim HcEq2: c = ((State m) 1) 1.
L182417
rewrite the current goal using HstEq (from left to right).
Use reflexivity.
L182418
rewrite the current goal using HcEq2 (from left to right).
Use reflexivity.
L182420
We prove the intermediate claim Hc_even_lt: ∀t : set, nat_p t → ((State (add_nat t t)) 1) 1 < eps_ t.
L182423
Apply nat_ind to the current goal.
L182424
We will prove ((State (add_nat 0 0)) 1) 1 < eps_ 0.
L182424
rewrite the current goal using (add_nat_0R 0) (from left to right) at position 1.
L182425
rewrite the current goal using eps_0_1 (from left to right).
L182426
An exact proof term for the current goal is (HInv_c_lt1 0 (nat_p_omega 0 nat_0)).
L182428
Let t be given.
L182428
Assume HtNat: nat_p t.
L182428
Assume IH: ((State (add_nat t t)) 1) 1 < eps_ t.
L182429
We will prove ((State (add_nat (ordsucc t) (ordsucc t))) 1) 1 < eps_ (ordsucc t).
L182430
Set N to be the term add_nat t t.
L182431
We prove the intermediate claim HNnat: nat_p N.
L182433
An exact proof term for the current goal is (add_nat_p t HtNat t HtNat).
L182433
We prove the intermediate claim Hidx: add_nat (ordsucc t) (ordsucc t) = ordsucc (ordsucc N).
L182435
rewrite the current goal using (add_nat_SL t HtNat (ordsucc t) (nat_ordsucc t HtNat)) (from left to right).
L182435
rewrite the current goal using (add_nat_SR t t HtNat) (from left to right).
Use reflexivity.
L182437
rewrite the current goal using Hidx (from left to right).
L182438
We prove the intermediate claim HN0: N ∈ ω.
L182440
An exact proof term for the current goal is (nat_p_omega N HNnat).
L182440
We prove the intermediate claim HcSN: ((State (ordsucc N)) 1) 1 = mul_SNo (((State N) 1) 1) den.
L182442
An exact proof term for the current goal is (Hc_step N HNnat).
L182442
We prove the intermediate claim HcSSN: ((State (ordsucc (ordsucc N))) 1) 1 = mul_SNo (mul_SNo (((State N) 1) 1) den) den.
L182445
We prove the intermediate claim HSNnat: nat_p (ordsucc N).
L182446
An exact proof term for the current goal is (nat_ordsucc N HNnat).
L182446
rewrite the current goal using (Hc_step (ordsucc N) HSNnat) (from left to right).
L182447
rewrite the current goal using HcSN (from left to right).
Use reflexivity.
L182449
rewrite the current goal using HcSSN (from left to right).
L182450
rewrite the current goal using (mul_SNo_assoc (((State N) 1) 1) den den (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))) HdenS HdenS) (from right to left).
L182453
We prove the intermediate claim Hden2Def: den2 = mul_SNo den den.
Use reflexivity.
L182455
rewrite the current goal using Hden2Def (from right to left).
L182456
We prove the intermediate claim HcNpos: 0 < ((State N) 1) 1.
L182458
An exact proof term for the current goal is (andER (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0)).
L182458
We prove the intermediate claim HcNS: SNo (((State N) 1) 1).
L182460
An exact proof term for the current goal is (real_SNo (((State N) 1) 1) (andEL (((State N) 1) 1 ∈ R) (0 < ((State N) 1) 1) (HInv_cpos N HN0))).
L182460
rewrite the current goal using (mul_SNo_com (((State N) 1) 1) den2 HcNS Hden2S) (from left to right).
L182461
We prove the intermediate claim Heps1S0: SNo (eps_ 1).
L182463
An exact proof term for the current goal is (real_SNo (eps_ 1) eps_1_in_R).
L182463
We prove the intermediate claim Heps1pos: 0 < eps_ 1.
L182465
An exact proof term for the current goal is (RltE_lt 0 (eps_ 1) eps_1_pos_R).
L182465
We prove the intermediate claim Hmul1: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (((State N) 1) 1).
L182467
An exact proof term for the current goal is (pos_mul_SNo_Lt' den2 (eps_ 1) (((State N) 1) 1) Hden2S Heps1S0 HcNS HcNpos Hden2Lt_eps1).
L182468
We prove the intermediate claim Heps_tR: (eps_ t) ∈ R.
L182470
An exact proof term for the current goal is (SNoS_omega_real (eps_ t) (SNo_eps_SNoS_omega t (nat_p_omega t HtNat))).
L182470
We prove the intermediate claim Heps_tS: SNo (eps_ t).
L182472
An exact proof term for the current goal is (real_SNo (eps_ t) Heps_tR).
L182472
We prove the intermediate claim Hmul2: mul_SNo (eps_ 1) (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L182474
We prove the intermediate claim HcomL: mul_SNo (eps_ 1) (((State N) 1) 1) = mul_SNo (((State N) 1) 1) (eps_ 1).
L182475
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (((State N) 1) 1) Heps1S0 HcNS).
L182475
We prove the intermediate claim HcomR: mul_SNo (eps_ 1) (eps_ t) = mul_SNo (eps_ t) (eps_ 1).
L182477
An exact proof term for the current goal is (mul_SNo_com (eps_ 1) (eps_ t) Heps1S0 Heps_tS).
L182477
rewrite the current goal using HcomL (from left to right).
L182478
rewrite the current goal using HcomR (from left to right).
L182479
An exact proof term for the current goal is (pos_mul_SNo_Lt' (((State N) 1) 1) (eps_ t) (eps_ 1) HcNS Heps_tS Heps1S0 Heps1pos IH).
L182481
We prove the intermediate claim HmulTra: mul_SNo den2 (((State N) 1) 1) < mul_SNo (eps_ 1) (eps_ t).
L182483
An exact proof term for the current goal is (SNoLt_tra (mul_SNo den2 (((State N) 1) 1)) (mul_SNo (eps_ 1) (((State N) 1) 1)) (mul_SNo (eps_ 1) (eps_ t)) (SNo_mul_SNo den2 (((State N) 1) 1) Hden2S HcNS) (SNo_mul_SNo (eps_ 1) (((State N) 1) 1) Heps1S0 HcNS) (SNo_mul_SNo (eps_ 1) (eps_ t) Heps1S0 Heps_tS) Hmul1 Hmul2).
L182489
We prove the intermediate claim HepsEq: mul_SNo (eps_ 1) (eps_ t) = eps_ (ordsucc t).
L182491
rewrite the current goal using (mul_SNo_eps_eps_add_SNo 1 (nat_p_omega 1 nat_1) t (nat_p_omega t HtNat)) (from left to right).
L182491
We prove the intermediate claim Hordt: ordinal t.
L182493
An exact proof term for the current goal is (nat_p_ordinal t HtNat).
L182493
rewrite the current goal using (ordinal_ordsucc_SNo_eq t Hordt) (from right to left).
Use reflexivity.
L182495
rewrite the current goal using HepsEq (from right to left).
L182496
An exact proof term for the current goal is HmulTra.
L182497
We prove the intermediate claim HcN0: ((State N0) 1) 1 < eps_ K.
L182499
An exact proof term for the current goal is (Hc_even_lt K HKnat).
L182499
We prove the intermediate claim HcN0R: ((State N0) 1) 1 ∈ R.
L182501
An exact proof term for the current goal is (andEL (((State N0) 1) 1 ∈ R) (0 < ((State N0) 1) 1) (HInv_cpos N0 HN0O)).
L182501
We prove the intermediate claim HcN0S: SNo (((State N0) 1) 1).
L182503
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcN0R).
L182503
We prove the intermediate claim HcN0ltEta: ((State N0) 1) 1 < eta.
L182505
An exact proof term for the current goal is (SNoLt_tra (((State N0) 1) 1) (eps_ K) eta HcN0S HepsKS HetaS HcN0 HepsKltEtaS).
L182505
We use N0 to witness the existential quantifier.
L182506
Apply andI to the current goal.
L182508
An exact proof term for the current goal is HN0O.
L182509
Let m and n be given.
L182509
Assume HmO: m ∈ ω.
L182509
Assume HnO: n ∈ ω.
L182509
Assume HNm: N0 ⊆ m.
L182510
Assume HNn: N0 ⊆ n.
L182510
Let x be given.
L182511
Assume HxX: x ∈ X.
L182511
We will prove Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps0.
L182512
We prove the intermediate claim HmNat: nat_p m.
L182514
An exact proof term for the current goal is (omega_nat_p m HmO).
L182514
We prove the intermediate claim HnNat: nat_p n.
L182516
An exact proof term for the current goal is (omega_nat_p n HnO).
L182516
We prove the intermediate claim Hexkm: ∃km : set, nat_p km ∧ m = add_nat km N0.
L182518
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat m HmNat HNm).
L182518
Apply Hexkm to the current goal.
L182519
Let km be given.
L182520
Assume Hkm.
L182520
We prove the intermediate claim HkmNat: nat_p km.
L182522
An exact proof term for the current goal is (andEL (nat_p km) (m = add_nat km N0) Hkm).
L182522
We prove the intermediate claim HmEq: m = add_nat km N0.
L182524
An exact proof term for the current goal is (andER (nat_p km) (m = add_nat km N0) Hkm).
L182524
rewrite the current goal using HmEq (from left to right).
L182525
We prove the intermediate claim Hexkn: ∃kn : set, nat_p kn ∧ n = add_nat kn N0.
L182527
An exact proof term for the current goal is (nat_Subq_add_ex N0 HN0Nat n HnNat HNn).
L182527
Apply Hexkn to the current goal.
L182528
Let kn be given.
L182529
Assume Hkn.
L182529
We prove the intermediate claim HknNat: nat_p kn.
L182531
An exact proof term for the current goal is (andEL (nat_p kn) (n = add_nat kn N0) Hkn).
L182531
We prove the intermediate claim HnEq: n = add_nat kn N0.
L182533
An exact proof term for the current goal is (andER (nat_p kn) (n = add_nat kn N0) Hkn).
L182533
rewrite the current goal using HnEq (from left to right).
L182534
We prove the intermediate claim HmO2: (add_nat km N0) ∈ ω.
L182536
An exact proof term for the current goal is (nat_p_omega (add_nat km N0) (add_nat_p km HkmNat N0 HN0Nat)).
L182536
We prove the intermediate claim HnO2: (add_nat kn N0) ∈ ω.
L182538
An exact proof term for the current goal is (nat_p_omega (add_nat kn N0) (add_nat_p kn HknNat N0 HN0Nat)).
L182538
We prove the intermediate claim HfnDef: fn = graph ω (λk : set ⇒ (State k) 0).
Use reflexivity.
L182540
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) (add_nat km N0) HfnDef HmO2) (from left to right).
L182541
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) (add_nat kn N0) HfnDef HnO2) (from left to right).
L182542
We prove the intermediate claim HmFS: (State (add_nat km N0)) 0 ∈ function_space X R.
L182544
An exact proof term for the current goal is (HInv_g_FS (add_nat km N0) HmO2).
L182544
We prove the intermediate claim HnFS: (State (add_nat kn N0)) 0 ∈ function_space X R.
L182546
An exact proof term for the current goal is (HInv_g_FS (add_nat kn N0) HnO2).
L182546
We prove the intermediate claim Hm_on: function_on ((State (add_nat km N0)) 0) X R.
L182548
An exact proof term for the current goal is (function_on_of_function_space ((State (add_nat km N0)) 0) X R HmFS).
L182548
We prove the intermediate claim Hn_on: function_on ((State (add_nat kn N0)) 0) X R.
L182550
An exact proof term for the current goal is (function_on_of_function_space ((State (add_nat kn N0)) 0) X R HnFS).
L182550
We prove the intermediate claim HmxR: apply_fun ((State (add_nat km N0)) 0) x ∈ R.
L182552
An exact proof term for the current goal is (Hm_on x HxX).
L182552
We prove the intermediate claim HnxR: apply_fun ((State (add_nat kn N0)) 0) x ∈ R.
L182554
An exact proof term for the current goal is (Hn_on x HxX).
L182554
We prove the intermediate claim HmxS: SNo (apply_fun ((State (add_nat km N0)) 0) x).
L182556
An exact proof term for the current goal is (real_SNo (apply_fun ((State (add_nat km N0)) 0) x) HmxR).
L182556
We prove the intermediate claim HnxS: SNo (apply_fun ((State (add_nat kn N0)) 0) x).
L182558
An exact proof term for the current goal is (real_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxR).
L182558
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L182560
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L182560
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L182562
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L182562
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x ∈ R.
L182564
An exact proof term for the current goal is (HgN0_on x HxX).
L182564
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x).
L182566
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x) HgN0xR).
L182566
We prove the intermediate claim HcN0S0: SNo (((State N0) 1) 1).
L182568
An exact proof term for the current goal is (real_SNo (((State N0) 1) 1) HcN0R).
L182568
We prove the intermediate claim Hstep_budget: ∀t : set, nat_p t → ∀x0 : set, x0 ∈ X → abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0))) ≤ add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
(*** step bound for the g component ***)
L182574
Let t be given.
L182574
Assume HtNat: nat_p t.
L182574
Let x0 be given.
L182575
Assume Hx0X: x0 ∈ X.
L182575
We prove the intermediate claim HtO: t ∈ ω.
L182577
An exact proof term for the current goal is (nat_p_omega t HtNat).
L182577
We prove the intermediate claim HtSuccO: ordsucc t ∈ ω.
L182579
An exact proof term for the current goal is (omega_ordsucc t HtO).
L182579
Set r to be the term ((State t) 1) 0.
L182580
Set c to be the term ((State t) 1) 1.
L182581
Set corr to be the term compose_fun X (u_of r) (mul_const_fun c).
L182582
Set gNew to be the term compose_fun X (pair_map X ((State t) 0) corr) add_fun_R.
L182583
Set cNew to be the term mul_SNo c den.
L182584
We prove the intermediate claim HS: State (ordsucc t) = StepState t (State t).
L182586
An exact proof term for the current goal is (nat_primrec_S BaseState StepState t HtNat).
L182586
rewrite the current goal using (Hc_step t HtNat) (from left to right) at position 1.
L182587
We prove the intermediate claim HgSucc: (State (ordsucc t)) 0 = gNew.
L182589
rewrite the current goal using HS (from left to right).
L182589
We prove the intermediate claim HdefS: StepState t (State t) = (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)).
Use reflexivity.
L182602
rewrite the current goal using HdefS (from left to right).
L182603
We prove the intermediate claim Hproj0: ((compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R,(compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)) 0) = (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R).
L182622
An exact proof term for the current goal is (tuple_2_0_eq (compose_fun X (pair_map X ((State t) 0) (compose_fun X (u_of (((State t) 1) 0)) (mul_const_fun (((State t) 1) 1)))) add_fun_R) (compose_fun A (compose_fun A (pair_map A (((State t) 1) 0) (compose_fun A (u_of (((State t) 1) 0)) neg_fun)) add_fun_R) (div_const_fun den),mul_SNo (((State t) 1) 1) den)).
L182633
rewrite the current goal using Hproj0 (from left to right).
L182634
We prove the intermediate claim HrEq: r = ((State t) 1) 0.
Use reflexivity.
L182636
We prove the intermediate claim HcEq0: c = ((State t) 1) 1.
Use reflexivity.
L182638
rewrite the current goal using HrEq (from right to left).
L182639
rewrite the current goal using HcEq0 (from right to left).
L182640
We prove the intermediate claim HcorrDef: corr = compose_fun X (u_of r) (mul_const_fun c).
Use reflexivity.
L182642
rewrite the current goal using HcorrDef (from right to left).
L182643
We prove the intermediate claim HgNewDef: gNew = compose_fun X (pair_map X ((State t) 0) corr) add_fun_R.
Use reflexivity.
L182645
rewrite the current goal using HgNewDef (from left to right).
Use reflexivity.
L182647
We prove the intermediate claim HcEq: c = ((State t) 1) 1.
Use reflexivity.
L182649
We prove the intermediate claim HcR: c ∈ R.
L182651
rewrite the current goal using HcEq (from left to right).
L182651
An exact proof term for the current goal is (andEL (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L182652
We prove the intermediate claim HcS: SNo c.
L182654
An exact proof term for the current goal is (real_SNo c HcR).
L182654
We prove the intermediate claim HcPos: 0 < c.
L182656
rewrite the current goal using HcEq (from left to right).
L182656
An exact proof term for the current goal is (andER (((State t) 1) 1 ∈ R) (0 < ((State t) 1) 1) (HInv_cpos t HtO)).
L182657
We prove the intermediate claim H0le_c: 0 ≤ c.
L182659
An exact proof term for the current goal is (SNoLtLe 0 c HcPos).
L182659
We prove the intermediate claim Hr_contI: continuous_map A Ta I Ti r.
L182661
An exact proof term for the current goal is (HInv_r_contI t HtO).
L182661
We prove the intermediate claim Hu_pack: continuous_map X Tx I0 T0 (u_of r) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x1 = one_third).
L182668
An exact proof term for the current goal is (Hu_of_prop r Hr_contI).
L182668
We prove the intermediate claim Hu_contI0: continuous_map X Tx I0 T0 (u_of r).
L182670
An exact proof term for the current goal is (andEL (continuous_map X Tx I0 T0 (u_of r)) (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third) (andEL ((continuous_map X Tx I0 T0 (u_of r)) ∧ (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval (minus_SNo 1) (minus_SNo one_third)) ∩ I) → apply_fun (u_of r) x1 = minus_SNo one_third)) (∀x1 : set, x1 ∈ preimage_of A r ((closed_interval one_third 1) ∩ I) → apply_fun (u_of r) x1 = one_third) Hu_pack)).
L182678
We prove the intermediate claim Hu_fun: function_on (u_of r) X I0.
L182680
An exact proof term for the current goal is (continuous_map_function_on X Tx I0 T0 (u_of r) Hu_contI0).
L182680
We prove the intermediate claim HuxI0: apply_fun (u_of r) x0 ∈ I0.
L182682
An exact proof term for the current goal is (Hu_fun x0 Hx0X).
L182682
We prove the intermediate claim HuxR: apply_fun (u_of r) x0 ∈ R.
L182684
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo one_third) one_third (apply_fun (u_of r) x0) HuxI0).
L182684
We prove the intermediate claim HuxS: SNo (apply_fun (u_of r) x0).
L182686
An exact proof term for the current goal is (real_SNo (apply_fun (u_of r) x0) HuxR).
L182686
We prove the intermediate claim HcorrEq: apply_fun corr x0 = mul_SNo (apply_fun (u_of r) x0) c.
L182689
We prove the intermediate claim HcorrDef0: corr = compose_fun X (u_of r) (mul_const_fun c).
Use reflexivity.
L182690
We prove the intermediate claim Hcomp: apply_fun corr x0 = apply_fun (mul_const_fun c) (apply_fun (u_of r) x0).
L182693
rewrite the current goal using HcorrDef0 (from left to right).
L182693
An exact proof term for the current goal is (compose_fun_apply X (u_of r) (mul_const_fun c) x0 Hx0X).
L182694
rewrite the current goal using Hcomp (from left to right).
L182695
An exact proof term for the current goal is (mul_const_fun_apply c (apply_fun (u_of r) x0) HcR HuxR).
L182696
We prove the intermediate claim HgxR: apply_fun ((State t) 0) x0 ∈ R.
L182698
An exact proof term for the current goal is (function_on_of_function_space ((State t) 0) X R (HInv_g_FS t HtO) x0 Hx0X).
L182698
We prove the intermediate claim HcorrR: apply_fun corr x0 ∈ R.
L182700
rewrite the current goal using HcorrEq (from left to right).
L182700
An exact proof term for the current goal is (real_mul_SNo (apply_fun (u_of r) x0) HuxR c HcR).
L182701
We prove the intermediate claim HgNewEval: apply_fun gNew x0 = add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0).
L182704
An exact proof term for the current goal is (add_of_pair_map_apply X ((State t) 0) corr x0 Hx0X HgxR HcorrR).
L182704
rewrite the current goal using HgSucc (from left to right).
L182705
rewrite the current goal using HgNewEval (from left to right).
L182706
We prove the intermediate claim Ha0S: SNo (apply_fun ((State t) 0) x0).
L182708
An exact proof term for the current goal is (real_SNo (apply_fun ((State t) 0) x0) HgxR).
L182708
We prove the intermediate claim Hb0S: SNo (apply_fun corr x0).
L182710
An exact proof term for the current goal is (real_SNo (apply_fun corr x0) HcorrR).
L182710
We prove the intermediate claim Hcancel: add_SNo (add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0)) (minus_SNo (apply_fun ((State t) 0) x0)) = apply_fun corr x0.
L182714
We prove the intermediate claim Hma0S: SNo (minus_SNo (apply_fun ((State t) 0) x0)).
L182715
An exact proof term for the current goal is (SNo_minus_SNo (apply_fun ((State t) 0) x0) Ha0S).
L182715
We prove the intermediate claim Hassoc1: add_SNo (apply_fun ((State t) 0) x0) (add_SNo (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0))) = add_SNo (add_SNo (apply_fun ((State t) 0) x0) (apply_fun corr x0)) (minus_SNo (apply_fun ((State t) 0) x0)).
L182722
An exact proof term for the current goal is (add_SNo_assoc (apply_fun ((State t) 0) x0) (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) Ha0S Hb0S Hma0S).
L182726
rewrite the current goal using Hassoc1 (from right to left) at position 1.
L182727
We prove the intermediate claim Hcom1: add_SNo (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) = add_SNo (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0).
L182731
An exact proof term for the current goal is (add_SNo_com (apply_fun corr x0) (minus_SNo (apply_fun ((State t) 0) x0)) Hb0S Hma0S).
L182731
rewrite the current goal using Hcom1 (from left to right) at position 1.
L182732
We prove the intermediate claim Hassoc2: add_SNo (apply_fun ((State t) 0) x0) (add_SNo (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0)) = add_SNo (add_SNo (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0))) (apply_fun corr x0).
L182739
An exact proof term for the current goal is (add_SNo_assoc (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0)) (apply_fun corr x0) Ha0S Hma0S Hb0S).
L182743
rewrite the current goal using Hassoc2 (from left to right).
L182744
We prove the intermediate claim Hinv: add_SNo (apply_fun ((State t) 0) x0) (minus_SNo (apply_fun ((State t) 0) x0)) = 0.
L182746
An exact proof term for the current goal is (add_SNo_minus_SNo_rinv (apply_fun ((State t) 0) x0) Ha0S).
L182746
rewrite the current goal using Hinv (from left to right) at position 1.
L182747
An exact proof term for the current goal is (add_SNo_0L (apply_fun corr x0) Hb0S).
L182748
rewrite the current goal using Hcancel (from left to right) at position 1.
L182749
rewrite the current goal using HcorrEq (from left to right).
L182750
Set u to be the term apply_fun (u_of r) x0.
L182751
We prove the intermediate claim HuAbsLe: abs_SNo u ≤ one_third.
L182753
We prove the intermediate claim Hm13R: minus_SNo one_third ∈ R.
L182754
An exact proof term for the current goal is (real_minus_SNo one_third one_third_in_R).
L182754
We prove the intermediate claim Hm13S: SNo (minus_SNo one_third).
L182756
An exact proof term for the current goal is (real_SNo (minus_SNo one_third) Hm13R).
L182756
We prove the intermediate claim H13S: SNo one_third.
L182758
An exact proof term for the current goal is (real_SNo one_third one_third_in_R).
L182758
We prove the intermediate claim Hbounds: Rle (minus_SNo one_third) u ∧ Rle u one_third.
L182760
An exact proof term for the current goal is (closed_interval_bounds (minus_SNo one_third) one_third u Hm13R one_third_in_R HuxI0).
L182760
We prove the intermediate claim Hlo: minus_SNo one_third ≤ u.
L182762
An exact proof term for the current goal is (SNoLe_of_Rle (minus_SNo one_third) u (andEL (Rle (minus_SNo one_third) u) (Rle u one_third) Hbounds)).
L182762
We prove the intermediate claim Hhi: u ≤ one_third.
L182764
An exact proof term for the current goal is (SNoLe_of_Rle u one_third (andER (Rle (minus_SNo one_third) u) (Rle u one_third) Hbounds)).
L182764
An exact proof term for the current goal is (abs_SNo_Le_of_bounds u one_third HuxS H13S Hlo Hhi).
L182765
We prove the intermediate claim Habsc: abs_SNo (mul_SNo u c) = mul_SNo (abs_SNo u) (abs_SNo c).
L182767
An exact proof term for the current goal is (abs_SNo_mul_eq u c HuxS HcS).
L182767
rewrite the current goal using Habsc (from left to right).
L182768
rewrite the current goal using (nonneg_abs_SNo c H0le_c) (from left to right).
L182769
We prove the intermediate claim H13S: SNo one_third.
L182771
An exact proof term for the current goal is (real_SNo one_third one_third_in_R).
L182771
We prove the intermediate claim HabsuS: SNo (abs_SNo u).
L182773
An exact proof term for the current goal is (SNo_abs_SNo u HuxS).
L182773
We prove the intermediate claim Hc13Le: mul_SNo (abs_SNo u) c ≤ mul_SNo one_third c.
L182775
An exact proof term for the current goal is (nonneg_mul_SNo_Le' (abs_SNo u) one_third c HabsuS H13S HcS H0le_c HuAbsLe).
L182775
We prove the intermediate claim HdenEq: den = two_thirds.
Use reflexivity.
L182777
rewrite the current goal using HdenEq (from left to right) at position 1.
L182778
We prove the intermediate claim H13Eq: mul_SNo one_third c = add_SNo c (minus_SNo (mul_SNo c two_thirds)).
L182780
We prove the intermediate claim H23S: SNo two_thirds.
L182781
An exact proof term for the current goal is (real_SNo two_thirds two_thirds_in_R).
L182781
We prove the intermediate claim Hc13S: SNo (mul_SNo c one_third).
L182783
An exact proof term for the current goal is (SNo_mul_SNo c one_third HcS H13S).
L182783
We prove the intermediate claim Hc23S: SNo (mul_SNo c two_thirds).
L182785
An exact proof term for the current goal is (SNo_mul_SNo c two_thirds HcS H23S).
L182785
rewrite the current goal using (mul_SNo_com one_third c H13S HcS) (from left to right).
L182786
We prove the intermediate claim HsumEq: add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds) = c.
L182788
rewrite the current goal using (mul_SNo_distrL c one_third two_thirds HcS H13S H23S) (from right to left).
L182788
rewrite the current goal using add_one_third_two_thirds_eq_1 (from left to right) at position 1.
L182789
An exact proof term for the current goal is (mul_SNo_oneR c HcS).
L182790
We prove the intermediate claim Hcan: add_SNo (add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds)) (minus_SNo (mul_SNo c two_thirds)) = mul_SNo c one_third.
L182793
An exact proof term for the current goal is (add_SNo_minus_R2 (mul_SNo c one_third) (mul_SNo c two_thirds) Hc13S Hc23S).
L182793
We prove the intermediate claim Htmp: add_SNo (add_SNo (mul_SNo c one_third) (mul_SNo c two_thirds)) (minus_SNo (mul_SNo c two_thirds)) = add_SNo c (minus_SNo (mul_SNo c two_thirds)).
L182796
rewrite the current goal using HsumEq (from left to right) at position 1.
Use reflexivity.
L182797
rewrite the current goal using Hcan (from right to left) at position 1.
L182798
rewrite the current goal using Htmp (from left to right) at position 1.
Use reflexivity.
L182800
rewrite the current goal using H13Eq (from right to left) at position 1.
L182801
An exact proof term for the current goal is Hc13Le.
L182802
We prove the intermediate claim Htail_budget: ∀k : set, nat_p k → ∀x0 : set, x0 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) ≤ ((State N0) 1) 1.
(*** tail budget from N0 via nat induction on shift ***)
L182808
Let k be given.
L182808
Assume HkNat: nat_p k.
L182808
Let x0 be given.
L182809
Assume Hx0X: x0 ∈ X.
L182809
We prove the intermediate claim Htail_strong_all: ∀kk : set, nat_p kk → ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat kk N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat kk N0)) 1) 1)).
L182814
Apply nat_ind to the current goal.
L182815
We will prove ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat 0 N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat 0 N0)) 1) 1)).
L182817
Let x1 be given.
L182818
Assume Hx1X: x1 ∈ X.
L182818
We prove the intermediate claim Hadd0: add_nat 0 N0 = N0.
L182820
An exact proof term for the current goal is (add_nat_0L N0 HN0Nat).
L182820
rewrite the current goal using Hadd0 (from left to right) at position 1.
L182821
rewrite the current goal using Hadd0 (from left to right) at position 1.
L182822
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L182824
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L182824
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L182826
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L182826
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x1 ∈ R.
L182828
An exact proof term for the current goal is (HgN0_on x1 Hx1X).
L182828
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x1).
L182830
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x1) HgN0xR).
L182830
rewrite the current goal using (add_SNo_minus_SNo_rinv (apply_fun ((State N0) 0) x1) HgN0xS) (from left to right) at position 1.
L182831
rewrite the current goal using (nonneg_abs_SNo 0 (SNoLe_ref 0)) (from left to right) at position 1.
L182832
rewrite the current goal using (add_SNo_minus_SNo_rinv (((State N0) 1) 1) HcN0S0) (from left to right) at position 1.
L182833
An exact proof term for the current goal is (SNoLe_ref 0).
L182835
Let kk be given.
L182835
Assume HkkNat: nat_p kk.
L182835
Assume IH: ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat kk N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat kk N0)) 1) 1)).
L182838
We will prove ∀x1 : set, x1 ∈ X → abs_SNo (add_SNo (apply_fun ((State (add_nat (ordsucc kk) N0)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat (ordsucc kk) N0)) 1) 1)).
L182841
Let x1 be given.
L182842
Assume Hx1X: x1 ∈ X.
L182842
Set t to be the term add_nat kk N0.
L182843
We prove the intermediate claim HtNat: nat_p t.
L182845
An exact proof term for the current goal is (add_nat_p kk HkkNat N0 HN0Nat).
L182845
We prove the intermediate claim HtO: t ∈ ω.
L182847
An exact proof term for the current goal is (nat_p_omega t HtNat).
L182847
We prove the intermediate claim Hidx: add_nat (ordsucc kk) N0 = ordsucc t.
L182849
rewrite the current goal using (add_nat_SL kk HkkNat N0 HN0Nat) (from left to right).
Use reflexivity.
L182850
rewrite the current goal using Hidx (from left to right) at position 1.
L182851
rewrite the current goal using Hidx (from left to right) at position 1.
L182852
We prove the intermediate claim HtSuccNat: nat_p (ordsucc t).
L182854
An exact proof term for the current goal is (nat_ordsucc t HtNat).
L182854
We prove the intermediate claim HtSuccO: ordsucc t ∈ ω.
L182856
An exact proof term for the current goal is (nat_p_omega (ordsucc t) HtSuccNat).
L182856
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L182858
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L182858
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L182860
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L182860
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x1 ∈ R.
L182862
An exact proof term for the current goal is (HgN0_on x1 Hx1X).
L182862
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x1).
L182864
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x1) HgN0xR).
L182864
We prove the intermediate claim HtFS: (State t) 0 ∈ function_space X R.
L182866
An exact proof term for the current goal is (HInv_g_FS t HtO).
L182866
We prove the intermediate claim Ht_on: function_on ((State t) 0) X R.
L182868
An exact proof term for the current goal is (function_on_of_function_space ((State t) 0) X R HtFS).
L182868
We prove the intermediate claim HtxR: apply_fun ((State t) 0) x1 ∈ R.
L182870
An exact proof term for the current goal is (Ht_on x1 Hx1X).
L182870
We prove the intermediate claim HtxS: SNo (apply_fun ((State t) 0) x1).
L182872
An exact proof term for the current goal is (real_SNo (apply_fun ((State t) 0) x1) HtxR).
L182872
We prove the intermediate claim HtSuccFS: (State (ordsucc t)) 0 ∈ function_space X R.
L182874
An exact proof term for the current goal is (HInv_g_FS (ordsucc t) HtSuccO).
L182874
We prove the intermediate claim HtSucc_on: function_on ((State (ordsucc t)) 0) X R.
L182876
An exact proof term for the current goal is (function_on_of_function_space ((State (ordsucc t)) 0) X R HtSuccFS).
L182876
We prove the intermediate claim HtSuccxR: apply_fun ((State (ordsucc t)) 0) x1 ∈ R.
L182878
An exact proof term for the current goal is (HtSucc_on x1 Hx1X).
L182878
We prove the intermediate claim HtSuccxS: SNo (apply_fun ((State (ordsucc t)) 0) x1).
L182880
An exact proof term for the current goal is (real_SNo (apply_fun ((State (ordsucc t)) 0) x1) HtSuccxR).
L182880
We prove the intermediate claim Htri: abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))).
L182886
An exact proof term for the current goal is (abs_SNo_triangle (apply_fun ((State (ordsucc t)) 0) x1) (apply_fun ((State t) 0) x1) (apply_fun ((State N0) 0) x1) HtSuccxS HtxS HgN0xS).
L182890
We prove the intermediate claim HstepLe: abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1))) ≤ add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
L182894
An exact proof term for the current goal is (Hstep_budget t HtNat x1 Hx1X).
L182894
We prove the intermediate claim HihLe: abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)).
L182898
An exact proof term for the current goal is (IH x1 Hx1X).
L182898
We prove the intermediate claim Habs1S: SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))).
L182900
An exact proof term for the current goal is (SNo_abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1))) (SNo_add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)) HtSuccxS (SNo_minus_SNo (apply_fun ((State t) 0) x1) HtxS))).
L182902
We prove the intermediate claim Habs2S: SNo (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))).
L182904
An exact proof term for the current goal is (SNo_abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) (SNo_add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)) HtxS (SNo_minus_SNo (apply_fun ((State N0) 0) x1) HgN0xS))).
L182906
We prove the intermediate claim HctR: (((State t) 1) 1) ∈ R.
L182908
An exact proof term for the current goal is (andEL ((((State t) 1) 1 ∈ R)) (0 < (((State t) 1) 1)) (HInv_cpos t HtO)).
L182908
We prove the intermediate claim HctS: SNo (((State t) 1) 1).
L182910
An exact proof term for the current goal is (real_SNo (((State t) 1) 1) HctR).
L182910
We prove the intermediate claim HctsR: (((State (ordsucc t)) 1) 1) ∈ R.
L182912
An exact proof term for the current goal is (andEL ((((State (ordsucc t)) 1) 1 ∈ R)) (0 < (((State (ordsucc t)) 1) 1)) (HInv_cpos (ordsucc t) HtSuccO)).
L182912
We prove the intermediate claim HctsS: SNo (((State (ordsucc t)) 1) 1).
L182914
An exact proof term for the current goal is (real_SNo (((State (ordsucc t)) 1) 1) HctsR).
L182914
We prove the intermediate claim Hc0S: SNo (((State N0) 1) 1).
L182916
An exact proof term for the current goal is HcN0S0.
L182916
We prove the intermediate claim HmctS: SNo (minus_SNo (((State t) 1) 1)).
L182918
An exact proof term for the current goal is (SNo_minus_SNo (((State t) 1) 1) HctS).
L182918
We prove the intermediate claim HmctsS: SNo (minus_SNo (((State (ordsucc t)) 1) 1)).
L182920
An exact proof term for the current goal is (SNo_minus_SNo (((State (ordsucc t)) 1) 1) HctsS).
L182920
We prove the intermediate claim Hrhs1S: SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))).
L182922
An exact proof term for the current goal is (SNo_add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)) HctS HmctsS).
L182922
We prove the intermediate claim Hrhs2S: SNo (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))).
L182924
An exact proof term for the current goal is (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)) Hc0S HmctS).
L182924
We prove the intermediate claim HsumLe: add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) ≤ add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))).
L182931
An exact proof term for the current goal is (add_SNo_Le3 (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Habs1S Habs2S Hrhs1S Hrhs2S HstepLe HihLe).
L182936
We prove the intermediate claim HsumEq: add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) = add_SNo (((State N0) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)).
L182942
rewrite the current goal using (add_SNo_com (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Hrhs1S Hrhs2S) (from left to right) at position 1.
L182945
rewrite the current goal using (add_SNo_com_4_inner_mid (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)) (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1)) Hc0S HmctS HctS HmctsS) (from left to right) at position 1.
L182951
We prove the intermediate claim Ht2S: SNo (add_SNo (((State N0) 1) 1) (((State t) 1) 1)).
L182953
An exact proof term for the current goal is (SNo_add_SNo (((State N0) 1) 1) (((State t) 1) 1) Hc0S HctS).
L182953
We prove the intermediate claim Ht3S: SNo (add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1))).
L182955
An exact proof term for the current goal is (SNo_add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1)) HmctS HmctsS).
L182955
rewrite the current goal using (add_SNo_assoc (((State N0) 1) 1) (((State t) 1) 1) (add_SNo (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1))) Hc0S HctS Ht3S) (from right to left) at position 1.
L182958
rewrite the current goal using (add_SNo_assoc (((State t) 1) 1) (minus_SNo (((State t) 1) 1)) (minus_SNo (((State (ordsucc t)) 1) 1)) HctS HmctS HmctsS) (from left to right) at position 1.
L182960
We prove the intermediate claim Hinv: add_SNo (((State t) 1) 1) (minus_SNo (((State t) 1) 1)) = 0.
L182962
An exact proof term for the current goal is (add_SNo_minus_SNo_rinv (((State t) 1) 1) HctS).
L182962
rewrite the current goal using Hinv (from left to right) at position 1.
L182963
rewrite the current goal using (add_SNo_0L (minus_SNo (((State (ordsucc t)) 1) 1)) HmctsS) (from left to right) at position 1.
Use reflexivity.
L182965
rewrite the current goal using HsumEq (from right to left) at position 1.
L182966
An exact proof term for the current goal is (SNoLe_tra (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) (add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))))) (add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1)))) (SNo_abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1))) (SNo_add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)) HtSuccxS (SNo_minus_SNo (apply_fun ((State N0) 0) x1) HgN0xS))) (SNo_add_SNo (abs_SNo (add_SNo (apply_fun ((State (ordsucc t)) 0) x1) (minus_SNo (apply_fun ((State t) 0) x1)))) (abs_SNo (add_SNo (apply_fun ((State t) 0) x1) (minus_SNo (apply_fun ((State N0) 0) x1)))) Habs1S Habs2S) (SNo_add_SNo (add_SNo (((State t) 1) 1) (minus_SNo (((State (ordsucc t)) 1) 1))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State t) 1) 1))) Hrhs1S Hrhs2S) Htri HsumLe).
L182985
We prove the intermediate claim Hstrong: abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) ≤ add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)).
L182989
An exact proof term for the current goal is (Htail_strong_all k HkNat x0 Hx0X).
L182989
We prove the intermediate claim HkN0Nat: nat_p (add_nat k N0).
L182991
An exact proof term for the current goal is (add_nat_p k HkNat N0 HN0Nat).
L182991
We prove the intermediate claim HkN0O: (add_nat k N0) ∈ ω.
L182993
An exact proof term for the current goal is (nat_p_omega (add_nat k N0) HkN0Nat).
L182993
We prove the intermediate claim HckR: (((State (add_nat k N0)) 1) 1) ∈ R.
L182995
An exact proof term for the current goal is (andEL ((((State (add_nat k N0)) 1) 1 ∈ R)) (0 < (((State (add_nat k N0)) 1) 1)) (HInv_cpos (add_nat k N0) HkN0O)).
L182995
We prove the intermediate claim HckS: SNo (((State (add_nat k N0)) 1) 1).
L182997
An exact proof term for the current goal is (real_SNo (((State (add_nat k N0)) 1) 1) HckR).
L182997
We prove the intermediate claim HckPos: 0 < (((State (add_nat k N0)) 1) 1).
L182999
An exact proof term for the current goal is (andER ((((State (add_nat k N0)) 1) 1 ∈ R)) (0 < (((State (add_nat k N0)) 1) 1)) (HInv_cpos (add_nat k N0) HkN0O)).
L182999
We prove the intermediate claim H0le_ck: 0 ≤ (((State (add_nat k N0)) 1) 1).
L183001
An exact proof term for the current goal is (SNoLtLe 0 (((State (add_nat k N0)) 1) 1) HckPos).
L183001
We prove the intermediate claim HmckLe0: minus_SNo (((State (add_nat k N0)) 1) 1) ≤ 0.
L183003
We prove the intermediate claim Htmp: minus_SNo (((State (add_nat k N0)) 1) 1) ≤ minus_SNo 0.
L183004
An exact proof term for the current goal is (minus_SNo_Le_contra 0 (((State (add_nat k N0)) 1) 1) SNo_0 HckS H0le_ck).
L183004
We prove the intermediate claim Hm0le0: minus_SNo 0 ≤ 0.
L183006
rewrite the current goal using minus_SNo_0 (from left to right).
L183006
An exact proof term for the current goal is (SNoLe_ref 0).
L183007
An exact proof term for the current goal is (SNoLe_tra (minus_SNo (((State (add_nat k N0)) 1) 1)) (minus_SNo 0) 0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS) (SNo_minus_SNo 0 SNo_0) SNo_0 Htmp Hm0le0).
L183015
We prove the intermediate claim HrhsLe: add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) ≤ ((State N0) 1) 1.
L183018
We prove the intermediate claim HsumLe0: add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) ≤ add_SNo (((State N0) 1) 1) 0.
L183021
An exact proof term for the current goal is (add_SNo_Le2 (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) 0 HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS) SNo_0 HmckLe0).
L183021
We prove the intermediate claim HmidLe: add_SNo (((State N0) 1) 1) 0 ≤ ((State N0) 1) 1.
L183024
rewrite the current goal using (add_SNo_0R (((State N0) 1) 1) HcN0S0) (from left to right).
L183024
An exact proof term for the current goal is (SNoLe_ref (((State N0) 1) 1)).
L183025
An exact proof term for the current goal is (SNoLe_tra (add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1))) (add_SNo (((State N0) 1) 1) 0) (((State N0) 1) 1) (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS)) (SNo_add_SNo (((State N0) 1) 1) 0 HcN0S0 SNo_0) HcN0S0 HsumLe0 HmidLe).
L183033
We prove the intermediate claim HgN0xS: SNo (apply_fun ((State N0) 0) x0).
L183035
We prove the intermediate claim HgN0FS: (State N0) 0 ∈ function_space X R.
L183036
An exact proof term for the current goal is (HInv_g_FS N0 HN0O).
L183036
We prove the intermediate claim HgN0_on: function_on ((State N0) 0) X R.
L183038
An exact proof term for the current goal is (function_on_of_function_space ((State N0) 0) X R HgN0FS).
L183038
We prove the intermediate claim HgN0xR: apply_fun ((State N0) 0) x0 ∈ R.
L183040
An exact proof term for the current goal is (HgN0_on x0 Hx0X).
L183040
An exact proof term for the current goal is (real_SNo (apply_fun ((State N0) 0) x0) HgN0xR).
L183041
An exact proof term for the current goal is (SNoLe_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0)))) (add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1))) (((State N0) 1) 1) (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0))) (SNo_add_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (minus_SNo (apply_fun ((State N0) 0) x0)) (real_SNo (apply_fun ((State (add_nat k N0)) 0) x0) (function_on_of_function_space ((State (add_nat k N0)) 0) X R (HInv_g_FS (add_nat k N0) HkN0O) x0 Hx0X)) (SNo_minus_SNo (apply_fun ((State N0) 0) x0) HgN0xS))) (SNo_add_SNo (((State N0) 1) 1) (minus_SNo (((State (add_nat k N0)) 1) 1)) HcN0S0 (SNo_minus_SNo (((State (add_nat k N0)) 1) 1) HckS)) HcN0S0 Hstrong HrhsLe).
L183053
We prove the intermediate claim Habs_mN0: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) < eta.
(*** derive abs difference < eps0 and conclude in bounded distance ***)
L183056
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (((State N0) 1) 1) eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) HcN0S0 HetaS (Htail_budget km HkmNat x HxX) HcN0ltEta).
L183063
We prove the intermediate claim HabSwap: abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) = abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))).
L183066
An exact proof term for the current goal is (abs_SNo_dist_swap (apply_fun ((State N0) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HgN0xS HnxS).
L183066
We prove the intermediate claim Habs_nN0: abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) < eta.
L183068
rewrite the current goal using HabSwap (from left to right).
L183068
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (((State N0) 1) 1) eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat kn N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HnxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) HcN0S0 HetaS (Htail_budget kn HknNat x HxX) HcN0ltEta).
L183076
We prove the intermediate claim HetaSumEq: add_SNo eta eta = eps0.
L183078
rewrite the current goal using (mul_SNo_distrL eps0 (eps_ 1) (eps_ 1) Heps0S Heps1S Heps1S) (from right to left).
L183078
rewrite the current goal using (eps_ordsucc_half_add 0 nat_0) (from left to right).
L183079
rewrite the current goal using eps_0_1 (from left to right).
L183080
rewrite the current goal using (mul_SNo_oneR eps0 Heps0S) (from left to right).
Use reflexivity.
L183082
We prove the intermediate claim HsumLt: add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) < eps0.
L183087
rewrite the current goal using HetaSumEq (from right to left).
L183087
An exact proof term for the current goal is (add_SNo_Lt3 (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) eta eta (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) (SNo_abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HgN0xS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS))) HetaS HetaS Habs_mN0 Habs_nN0).
L183097
We prove the intermediate claim HtriMN: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) ≤ add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))).
L183102
An exact proof term for the current goal is (abs_SNo_triangle (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State N0) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HmxS HgN0xS HnxS).
L183103
We prove the intermediate claim HabsLt: abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) < eps0.
L183105
An exact proof term for the current goal is (SNoLeLt_tra (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) (add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))))) eps0 (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS))) (SNo_add_SNo (abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)))) (abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)))) (SNo_abs_SNo (add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x))) (SNo_add_SNo (apply_fun ((State (add_nat km N0)) 0) x) (minus_SNo (apply_fun ((State N0) 0) x)) HmxS (SNo_minus_SNo (apply_fun ((State N0) 0) x) HgN0xS))) (SNo_abs_SNo (add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x))) (SNo_add_SNo (apply_fun ((State N0) 0) x) (minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x)) HgN0xS (SNo_minus_SNo (apply_fun ((State (add_nat kn N0)) 0) x) HnxS)))) Heps0S HtriMN HsumLt).
L183123
We prove the intermediate claim Hpair: (apply_fun ((State (add_nat km N0)) 0) x,apply_fun ((State (add_nat kn N0)) 0) x) ∈ setprod R R.
L183125
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) HmxR HnxR).
L183125
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L183128
rewrite the current goal using HdefM (from left to right).
L183129
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun ((State (add_nat km N0)) 0) x,apply_fun ((State (add_nat kn N0)) 0) x) Hpair) (from left to right).
L183131
rewrite the current goal using (tuple_2_0_eq (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x)) (from left to right).
L183132
rewrite the current goal using (tuple_2_1_eq (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x)) (from left to right).
L183133
An exact proof term for the current goal is (abs_lt_lt1_imp_R_bounded_distance_lt (apply_fun ((State (add_nat km N0)) 0) x) (apply_fun ((State (add_nat kn N0)) 0) x) eps0 HmxR HnxR Heps0R Heps0Lt1 HabsLt).
L183138
Apply (SNoLt_trichotomy_or_impred eps 1 HepsS SNo_1 (∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps)) to the current goal.
L183144
Assume HepsLt1S: eps < 1.
L183144
We prove the intermediate claim HepsLt1: Rlt eps 1.
L183146
An exact proof term for the current goal is (RltI eps 1 HepsR real_1 HepsLt1S).
L183146
An exact proof term for the current goal is (Huc_small eps HepsR HepsPos HepsLt1).
L183148
Assume HepsEq1: eps = 1.
L183148
We prove the intermediate claim HexN: ∃N : set, N ∈ ω ∧ ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L183153
An exact proof term for the current goal is (Huc_small (eps_ 1) eps_1_in_R eps_1_pos_R eps_1_lt1_R).
L183153
Apply HexN to the current goal.
L183154
Let N be given.
L183155
Assume HN.
L183155
We use N to witness the existential quantifier.
L183156
Apply andI to the current goal.
L183158
An exact proof term for the current goal is (andEL (N ∈ ω) (∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1)) HN).
L183163
Let m and n be given.
L183163
Assume HmO: m ∈ ω.
L183163
Assume HnO: n ∈ ω.
L183163
Assume HNm: N ⊆ m.
L183164
Assume HNn: N ⊆ n.
L183164
Let x be given.
L183165
Assume HxX: x ∈ X.
L183165
We prove the intermediate claim HNprop: ∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L183170
An exact proof term for the current goal is (andER (N ∈ ω) (∀m n : set, m ∈ ω → n ∈ ω → N ⊆ m → N ⊆ n → ∀x : set, x ∈ X → Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1)) HN).
L183174
We prove the intermediate claim Hlt1: Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1).
L183176
An exact proof term for the current goal is (HNprop m n HmO HnO HNm HNn x HxX).
L183176
We prove the intermediate claim Hlt2: Rlt (eps_ 1) eps.
L183178
rewrite the current goal using HepsEq1 (from left to right).
L183178
An exact proof term for the current goal is eps_1_lt1_R.
L183179
An exact proof term for the current goal is (Rlt_tra (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) (eps_ 1) eps Hlt1 Hlt2).
L183181
Assume H1LtEpsS: 1 < eps.
L183181
We prove the intermediate claim H1LtEps: Rlt 1 eps.
L183183
An exact proof term for the current goal is (RltI 1 eps real_1 HepsR H1LtEpsS).
L183183
We use 0 to witness the existential quantifier.
L183184
Apply andI to the current goal.
L183186
An exact proof term for the current goal is (nat_p_omega 0 nat_0).
L183187
Let m and n be given.
L183187
Assume HmO: m ∈ ω.
L183187
Assume HnO: n ∈ ω.
L183187
Assume HNm: 0 ⊆ m.
L183188
Assume HNn: 0 ⊆ n.
L183188
Let x be given.
L183189
Assume HxX: x ∈ X.
L183189
We will prove Rlt (apply_fun R_bounded_metric (apply_fun (apply_fun fn m) x,apply_fun (apply_fun fn n) x)) eps.
L183190
We prove the intermediate claim HfnDef: fn = graph ω (λk : set ⇒ (State k) 0).
Use reflexivity.
L183192
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) m HfnDef HmO) (from left to right).
L183193
rewrite the current goal using (apply_fun_of_graph_eq fn ω (λk : set ⇒ (State k) 0) n HfnDef HnO) (from left to right).
L183194
We prove the intermediate claim HmFS: (State m) 0 ∈ function_space X R.
L183196
An exact proof term for the current goal is (HInv_g_FS m HmO).
L183196
We prove the intermediate claim HnFS: (State n) 0 ∈ function_space X R.
L183198
An exact proof term for the current goal is (HInv_g_FS n HnO).
L183198
We prove the intermediate claim Hm_on: function_on ((State m) 0) X R.
L183200
An exact proof term for the current goal is (function_on_of_function_space ((State m) 0) X R HmFS).
L183200
We prove the intermediate claim Hn_on: function_on ((State n) 0) X R.
L183202
An exact proof term for the current goal is (function_on_of_function_space ((State n) 0) X R HnFS).
L183202
We prove the intermediate claim HmxR: apply_fun ((State m) 0) x ∈ R.
L183204
An exact proof term for the current goal is (Hm_on x HxX).
L183204
We prove the intermediate claim HnxR: apply_fun ((State n) 0) x ∈ R.
L183206
An exact proof term for the current goal is (Hn_on x HxX).
L183206
We prove the intermediate claim Hpair: (apply_fun ((State m) 0) x,apply_fun ((State n) 0) x) ∈ setprod R R.
L183208
An exact proof term for the current goal is (tuple_2_setprod_by_pair_Sigma R R (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x) HmxR HnxR).
L183208
We prove the intermediate claim HdefM: R_bounded_metric = graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)).
Use reflexivity.
L183211
rewrite the current goal using HdefM (from left to right).
L183212
rewrite the current goal using (apply_fun_graph (setprod R R) (λp : set ⇒ R_bounded_distance (p 0) (p 1)) (apply_fun ((State m) 0) x,apply_fun ((State n) 0) x) Hpair) (from left to right).
L183214
rewrite the current goal using (tuple_2_0_eq (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) (from left to right).
L183215
rewrite the current goal using (tuple_2_1_eq (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) (from left to right).
L183216
We prove the intermediate claim Hle1: Rle (R_bounded_distance (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) 1.
L183218
An exact proof term for the current goal is (R_bounded_distance_le_1 (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x) HmxR HnxR).
L183218
An exact proof term for the current goal is (Rle_Rlt_tra (R_bounded_distance (apply_fun ((State m) 0) x) (apply_fun ((State n) 0) x)) 1 eps Hle1 H1LtEps).
L183219
Apply Hexfn to the current goal.
L183220
Let fn be given.
L183221
Assume Hfnpack: function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) ∧ uniform_cauchy_metric X R R_bounded_metric fn.
L183229
We prove the intermediate claim Hfn1234: ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)).
L183240
An exact proof term for the current goal is (andEL (((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x))) (uniform_cauchy_metric X R R_bounded_metric fn) Hfnpack).
L183251
We prove the intermediate claim Hfn3: uniform_cauchy_metric X R R_bounded_metric fn.
L183253
An exact proof term for the current goal is (andER ((((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) ∧ (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)))) (uniform_cauchy_metric X R R_bounded_metric fn) Hfnpack).
L183264
We prove the intermediate claim Hfn123: (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I).
L183270
An exact proof term for the current goal is (andEL ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) Hfn1234).
L183279
We prove the intermediate claim HfnLimA: ∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x).
L183285
An exact proof term for the current goal is (andER ((function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) ∧ (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I)) (∀x : set, x ∈ A → converges_to R (metric_topology R R_bounded_metric) (graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x)) (apply_fun f x)) Hfn1234).
L183294
We prove the intermediate claim Hfn12: function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)).
L183298
An exact proof term for the current goal is (andEL (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) Hfn123).
L183302
We prove the intermediate claim HfnRange: ∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I.
L183305
An exact proof term for the current goal is (andER (function_on fn ω (function_space X R) ∧ (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n))) (∀n : set, n ∈ ω → ∀x : set, x ∈ X → apply_fun (apply_fun fn n) x ∈ I) Hfn123).
L183309
We prove the intermediate claim Hfn1: function_on fn ω (function_space X R).
L183311
An exact proof term for the current goal is (andEL (function_on fn ω (function_space X R)) (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) Hfn12).
L183314
We prove the intermediate claim Hfn2: ∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n).
L183316
An exact proof term for the current goal is (andER (function_on fn ω (function_space X R)) (∀n : set, n ∈ ω → continuous_map X Tx R R_standard_topology (apply_fun fn n)) Hfn12).
L183319
We prove the intermediate claim HexgR: ∃gR : set, function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR ∧ continuous_map X Tx R R_standard_topology gR.
L183324
An exact proof term for the current goal is (uniform_cauchy_continuous_to_R_has_continuous_limit X Tx fn HTx Hfn1 Hfn2 Hfn3).
L183329
Apply HexgR to the current goal.
L183330
Let gR be given.
L183331
Assume HgRpack: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR ∧ continuous_map X Tx R R_standard_topology gR.
L183334
We use gR to witness the existential quantifier.
L183335
Apply andI to the current goal.
L183337
Apply andI to the current goal.
L183338
An exact proof term for the current goal is (andER (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L183341
Let x be given.
L183341
Assume HxA: x ∈ A.
L183341
We will prove apply_fun gR x = apply_fun f x.
L183342
Apply (xm (apply_fun gR x = apply_fun f x)) to the current goal.
L183344
Assume Heq: apply_fun gR x = apply_fun f x.
L183344
An exact proof term for the current goal is Heq.
L183346
Assume Hneq: ¬ (apply_fun gR x = apply_fun f x).
L183346
We will prove apply_fun gR x = apply_fun f x.
L183347
Apply FalseE to the current goal.
L183348
We prove the intermediate claim HxX': x ∈ X.
L183350
An exact proof term for the current goal is (HAsubX x HxA).
L183350
We prove the intermediate claim HmR: metric_on R R_bounded_metric.
L183352
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L183352
We prove the intermediate claim HHaus: Hausdorff_space R (metric_topology R R_bounded_metric).
L183354
An exact proof term for the current goal is (metric_topology_Hausdorff R R_bounded_metric HmR).
L183354
We prove the intermediate claim HgR12: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR.
L183356
An exact proof term for the current goal is (andEL (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L183359
We prove the intermediate claim HgRfun: function_on gR X R.
L183361
An exact proof term for the current goal is (andEL (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L183364
We prove the intermediate claim HunifRgR: uniform_limit_metric X R R_bounded_metric fn gR.
L183366
An exact proof term for the current goal is (andER (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L183369
Set seqx to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L183370
We prove the intermediate claim Hxlim_g: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun gR x).
L183372
An exact proof term for the current goal is (uniform_limit_metric_imp_converges_to_metric_topology_at_point X R R_bounded_metric fn gR x HmR Hfn1 HgRfun HxX' HunifRgR).
L183374
We prove the intermediate claim Hxlim_f: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun f x).
L183376
An exact proof term for the current goal is (HfnLimA x HxA).
L183376
We prove the intermediate claim HgxR: apply_fun gR x ∈ R.
L183378
An exact proof term for the current goal is (HgRfun x HxX').
L183378
We prove the intermediate claim HfxR: apply_fun f x ∈ R.
L183380
An exact proof term for the current goal is (Hf_R x HxA).
L183380
We prove the intermediate claim Hneqxy: apply_fun gR x ≠ apply_fun f x.
L183382
Assume Heq: apply_fun gR x = apply_fun f x.
L183382
An exact proof term for the current goal is (Hneq Heq).
L183383
We prove the intermediate claim Hseq_on: function_on seqx ω R.
L183385
Let n be given.
L183385
Assume HnO: n ∈ ω.
L183385
We will prove apply_fun seqx n ∈ R.
L183386
We prove the intermediate claim Hseqxdef: seqx = graph ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x).
Use reflexivity.
L183388
rewrite the current goal using (apply_fun_of_graph_eq seqx ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x) n Hseqxdef HnO) (from left to right).
L183389
We prove the intermediate claim HfnnxI: apply_fun (apply_fun fn n) x ∈ I.
L183391
An exact proof term for the current goal is (HfnRange n HnO x HxX').
L183391
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1 (apply_fun (apply_fun fn n) x) HfnnxI).
L183392
We prove the intermediate claim Hnbhd_g: ∀U : set, U ∈ metric_topology R R_bounded_metric → apply_fun gR x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seqx n ∈ U.
L183396
An exact proof term for the current goal is (converges_to_neighborhoods R (metric_topology R R_bounded_metric) seqx (apply_fun gR x) Hxlim_g).
L183398
We prove the intermediate claim Hnbhd_f: ∀U : set, U ∈ metric_topology R R_bounded_metric → apply_fun f x ∈ U → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seqx n ∈ U.
L183402
An exact proof term for the current goal is (converges_to_neighborhoods R (metric_topology R R_bounded_metric) seqx (apply_fun f x) Hxlim_f).
L183404
An exact proof term for the current goal is (Hausdorff_unique_limits R (metric_topology R R_bounded_metric) seqx (apply_fun gR x) (apply_fun f x) HHaus HgxR HfxR Hneqxy Hseq_on Hnbhd_g Hnbhd_f).
L183414
Let x be given.
L183414
Assume HxX: x ∈ X.
L183414
We will prove apply_fun gR x ∈ I.
L183415
We prove the intermediate claim HmR: metric_on R R_bounded_metric.
L183417
An exact proof term for the current goal is R_bounded_metric_is_metric_on_early.
L183417
We prove the intermediate claim HTmR: topology_on R (metric_topology R R_bounded_metric).
L183419
An exact proof term for the current goal is (metric_topology_is_topology R R_bounded_metric HmR).
L183419
We prove the intermediate claim HI_sub_R: I ⊆ R.
L183421
An exact proof term for the current goal is (closed_interval_sub_R (minus_SNo 1) 1).
L183421
We prove the intermediate claim Hm1R': (minus_SNo 1) ∈ R.
L183423
An exact proof term for the current goal is (real_minus_SNo 1 real_1).
L183423
We prove the intermediate claim HI_closed: closed_in R (metric_topology R R_bounded_metric) I.
L183425
rewrite the current goal using metric_topology_R_bounded_metric_eq_R_standard_topology_early (from left to right).
L183425
An exact proof term for the current goal is (closed_interval_closed_in_R_standard_topology (minus_SNo 1) 1 Hm1R' real_1).
L183426
We prove the intermediate claim HgR12: function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR.
L183428
An exact proof term for the current goal is (andEL (function_on gR X R ∧ uniform_limit_metric X R R_bounded_metric fn gR) (continuous_map X Tx R R_standard_topology gR) HgRpack).
L183431
We prove the intermediate claim HgRfun: function_on gR X R.
L183433
An exact proof term for the current goal is (andEL (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L183436
We prove the intermediate claim HunifRgR: uniform_limit_metric X R R_bounded_metric fn gR.
L183438
An exact proof term for the current goal is (andER (function_on gR X R) (uniform_limit_metric X R R_bounded_metric fn gR) HgR12).
L183441
Set seqx to be the term graph ω (λn : set ⇒ apply_fun (apply_fun fn n) x).
L183442
We prove the intermediate claim Hconvx: converges_to R (metric_topology R R_bounded_metric) seqx (apply_fun gR x).
L183444
An exact proof term for the current goal is (uniform_limit_metric_imp_converges_to_metric_topology_at_point X R R_bounded_metric fn gR x HmR Hfn1 HgRfun HxX HunifRgR).
L183446
We prove the intermediate claim HseqxI: ∀n : set, n ∈ ω → apply_fun seqx n ∈ I.
L183448
Let n be given.
L183448
Assume HnO: n ∈ ω.
L183448
We prove the intermediate claim Hseqxdef: seqx = graph ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x).
Use reflexivity.
L183450
rewrite the current goal using (apply_fun_of_graph_eq seqx ω (λn0 : set ⇒ apply_fun (apply_fun fn n0) x) n Hseqxdef HnO) (from left to right).
L183451
An exact proof term for the current goal is (HfnRange n HnO x HxX).
L183452
An exact proof term for the current goal is (converges_to_closed_in_contains_limit R (metric_topology R R_bounded_metric) I seqx (apply_fun gR x) HTmR HI_sub_R HI_closed Hconvx HseqxI).
L183459
An exact proof term for the current goal is Hseries.
∎